Brauer groups and Galois cohomology of function fields of varieties

Jason Michael Starr

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794

E-mail address: jstarr@math.sunysb.edu

Contents

1. Acknowledgments  5
2. Introduction  7
Chapter 1. Brauer groups and Galois cohomology  9
1. Abelian Galois cohomology  9
2. Non-Abelian Galois cohomology and the long exact sequence  13
3. Galois cohomology of smooth group schemes  22
4. The Brauer group  29
5. The universal cover sequence  34
Chapter 2. The Chevalley-Warning and Tsen-Lang theorems  37
1. The Chevalley-Warning Theorem  37
2. The Tsen-Lang Theorem  39
3. Applications to Brauer groups  43
Chapter 3. Rationally connected fibrations over curves  47
1. Rationally connected varieties  47
2. Outline of the proof  51
3. Hilbert schemes and smoothing combs  54
4. Ramification issues  63
5. Existence of log deformations  68
6. Completion of the proof  70
7. Corollaries  72
Chapter 4. The Period-Index theorem of de Jong  75
1. Statement of the theorem  75
2. Abel maps over curves and sections over surfaces  78
3. Rational simple connectedness hypotheses  79
4. Rational connectedness of the Abel map  81
5. Rational simply connected fibrations over a surface  82
6. Discriminant avoidance  84
7. Proof of the main theorem for Grassmann bundles  86
Chapter 5. Rational simple connectedness and Serre’s “Conjecture II”  89
1. Generalized Grassmannians are rationally simply connected  89
2. Statement of the theorem  90
3. Reductions of structure group  90
Bibliography  93

4

1. Acknowledgments

Chapters 2 and 3 notes are largely adapted from notes for a similar lecture series presented at the Clay Mathematics Institute Summer School in Göttingen, Germany in Summer 2006. I am grateful to the Clay Mathematics Institute and the Universität Göttingen for sponsoring that school.

Chapters 4 and 5 are largely adapted from notes for a lecture series presented at the Etats de la Recherche 2008 de la Société Mathématique de France conference Variétés rationnellement connexes : aspects géométriques et arithmétiques presented at the Université Louis Pasteur Strasbourg in May 2008. I am grateful to the Société Mathématique de France and the CNRS for sponsoring that school.

I am grateful to Brendan Hassett, Yuri Tschinkel and A. J. de Jong for useful conversations on the content and exposition of these notes. These notes report on joint work with Joe Harris and Tom Graber, as well as joint work with A. J. de Jong and Xuhua He. Moreover, the results in Chapter 5 are deeply motivated by a strategy of Philippe Gille for proving the split case of Serre’s “Conjecture II” which he explained to us some years ago. I express my deepest gratitude to Gille for showing us that rational simple connectedness is related to Serre’s conjecture.

I am partially supported by grants from the National Science Foundation as well as an Alfred P. Sloan research fellowship. Finally, I very gratefully acknowledge the support of INSTITUTO NACIONAL DE MATEMÁTICA PURA E APLICADA and the Brazilian government for making the XX Escola de Algebra possible.

6

2. Introduction

These are the lecture notes for a minicourse presented in the XX Escola de Algebra at the Instituto Nacional de Matemática Pura e Aplicada in Rio de Janeiro, Brazil, in August of 2008. The minicourse begins with an introduction to Brauer groups and Galois cohomology followed by the proofs of the Chevalley-Warning and Tsen-Lang theorems and their applications to Brauer groups and Galois cohomology. The minicourse concludes with some more recent theorems about Brauer groups and Galois cohomology of function fields of varieties over an algebraically closed field: de Jong’s Period-Index theorem and the split case of Serre’s ”Conjecture II” over function fields of surfaces due to de Jong, He and myself and following a strategy proposed by Philippe Gille. The split case completes the proof of the general case of Serre’s ”Conjecture II” over function fields of surfaces following the work of Merkurjev, Suslin, Bayer-Fluckiger, Parimala, Colliot-Thélène, and Gille.

In addition to introducing students and post-docs to the classical subjects of Brauer groups and Galois cohomology, a second goal of the minicourse is to show by example how results from geometry – specifically the geometry of rationally connected varieties and the geometry of spaces of curves – can be used to prove results in algebra – namely the Period-Index theorem and the split case of Serre’s ”Conjecture II”.

Here is a short description of the contents of the minicourse.

Chapter 1: Brauer groups and Galois cohomology. Definition of division algebras and central simple algebras. Definition of the Brauer group. Short sketch of Galois cohomology and its relation to the Brauer group.

Chapter 2: The Chevalley-Warning and Tsen-Lang theorems. The statement and proofs of Chevalley’s theorem and Tsen’s theorem which prove that the Brauer group of a finite field, resp. the function field of a curve over an algebraically closed field, is trivial. Statement of the generalized results of Chevalley-Warning and Tsen-Lang with applications to Brauer groups and Galois cohomology.

Chapter 3: Rationally connected fibrations. A sketch of rationally connected varieties. The statement of the conjecture of Kollár-Miyaoka-Mori that every rationally connected fibration over a curve has a section. A sketch of the proof of this conjecture by Graber, Harris and myself.

Chapter 4: The Period-Index theorem of de Jong Explanation of de Jong’s Period-Index theorem: for every division algebra over the function field of a surface, the period equals the index. Sketch of a second proof, by de Jong and myself, using rational simple connectedness. Comparison to a third proof by Lieblich, which also proves generalizations of de Jong’s theorem.

Chapter 5: Rational simple connectedness and Serre’s ”Conjecture II” Description of known results about Serre’s “Conjecture II”: theorems of Merkurjev and Suslin, Bayer-Fluckiger and Parimala, Colliot-Thélène, Gille and Parimala, and Gille. Sketch of the proof of the split case of Serre’s ”Conjecture II” over function fields of surfaces using rational simple connectedness.

CHAPTER 1 Brauer groups and Galois cohomology

1. Abelian Galois cohomology

The first lecture gives a quick survey of some results in Galois cohomology and Brauer groups. Most results are stated without proof or with only an outline of a proof. The main references for this lecture are [Ser02] and [Ser79]. For results about homological algebra, the reference is [Wei94].

Let L/K be a Galois extension of fields, e.g., a separable closure K^{s} of K. The Galois group Gal(L/K) is a profinite group, namely the inverse limit of the Galois groups Gal(L_{i}/K) of all finite Galois subextensions L_{i}, i.e., K ⊂ L_{i} ⊂ L.

Definition 1.1. An Abelian Gal(L/K)-module is an Abelian group A together with a left action of Gal(L/K) by group homomorphisms. The module is discrete if every element of A is stabilized by Gal(L_{i}/K) for some finite Galois subextension L_{i}/K, or equivalently, if the canonical map of Abelian Gal(L/K)-modules

A → lim_{K⊂L_{i}⊂L} A_{i}, A_{i} := A^{Gal(L/L_{i})},

is a bijection.

The Abelian category of all Abelian Gal(L/K)-modules is denoted Gal(L/K)− mod. And the full Abelian subcategory of discrete Abelian Gal(L/K)-modules is denoted Gal(L/K) − mod_{discrete}.

Let L_{i}/K be a finite Galois subextension of L/K. There is a left-exact additive functor

H^{0}(Gal(L_{i}/K), −) : Gal(L_{i}/K) − mod → ℤ − mod,
B ↦ B^{Gal(L_{i}/K)},

the functor of Gal(L_{i}/K)-invariants.

Definition 1.2. The sequence of right derived functors of H^{0}(Gal(L_{i}/K), −) considered as a cohomological δ-functor is the Galois cohomology or Gal(L_{i}/K)- cohomology,

H^{p}(Gal(L_{i}/K), −) : Gal(L_{i}/K) − mod → ℤ − mod,
H^{p}(Gal(L_{i}/K), B) := R^{p}H^{0}(Gal(L_{i}/K), B).

For a discrete Abelian Gal(L/K)-module A, for every integer p ≥ 0 the sequence of group cohomologies (H^{p}(Gal(L_{i}/K), A_{i}))_{L_{i}} form a directed system of Abelian groups. Similarly, for every exact sequence of discrete Abelian Gal(L/K)-modules,

0 ⟶ A′ ⟶ A ⟶ A′′ ⟶ 0,

the collection of connecting maps

δ_{L_{i}}^{p} : H^{p}(Gal(L_{i}/K), A′′_{i}) → H^{p+1}(Gal(L_{i}/K), A′_{i})

is a morphism of compatible systems.

Definition 1.3. The Galois cohomology or Gal(L/K)-cohomology is the cohomological δ-functor from Gal(L/K)−mod_{discrete} to ℤ−mod defined by the colimits

H^{p}(Gal(L/K), −) : Gal(L/K) − mod_{discrete} → ℤ − mod,

H^{p}(Gal(L/K), A) := lim_{L_{i}} H^{p}(Gal(L_{i}/K), A_{i}),

and

δ^{p} : H^{p}(Gal(L/K), A′′) → H^{p+1}(Gal(L/K), A′), δ_{p} = lim_{L_{i}} δ_{L_{i}}^{p}.

This is a universal δ-functor. In fact the category Gal(L/K) − mod_{discrete} has enough injective objects, and one can prove that the usual sequence of right derived functors of A^{Gal(L/K)} is canonically isomorphic as a δ-functor to the one from Definition 1.3. Definition 1.3 reflects one technique of studying Gal(L/K)- cohomology: reduction to the case of finite Galois extensions L_{i}/K.

For the reader who knows of such things, Gal(L/K)-cohomology is nothing other than continuous group cohomology for the group Gal(L/K) with its profinite topology. This raises the question, how is the study of Galois cohomology different from the study of profinite group cohomology? The answer has to do mainly with the particular Galois modules of interest.

1.1. Low degree Galois cohomology. The low degree Galois cohomology groups have interpretations of special interest.

1.1.1. The zeroth group. Let L/K be a Galois extension and let A be a discrete Gal(L/K)-module.

Proposition 1.4. There is a canonical bijection between H^{0}(Gal(L/K), A) and the normal subgroup A^{Gal(L/K)} of elements of A left invariant under Gal(L/K).

Proof. By definition, for every finite, Galois subextension K ⊂ L_{i} ⊂ L

H^{0}(Gal(L_{i}/K), A_{i}) = A_{i}^{Gal(L_{i}/K)}.

Since A_{i} is defined to be A^{Gal(L/L_{i})}, this gives

H^{0}(Gal(L_{i}/K), A_{i}) = A^{Gal(L/K)}.

Thus also

H^{0}(Gal(L/K), A) = A^{Gal(L/K)},

i.e., H^{0}(Gal(L/K), A) equals the subgroup of elements of A left invariant under Gal(L/K). □

1.1.2. The first group, Interpretation I.. Let L/K be a Galois extension and let A be a discrete, Abelian Gal(L/K)-module. A crossed homomorphism from Gal(L/K) into A is a set map

a : Gal(L/K) → A

such that for every g, h ∈ Gal(L/K),

a_{g·h} = a_{g} ·^{g}a_{h}.

The crossed homomorphism is continuous if it factors through Gal(L_{i}/K) for some finite subextension K ⊂ L_{i} ⊂ L.

There is a distinguished continuous crossed homomorphism e, namely

e_{g} = e,

the identity element of A for every g in Gal(L_{i}/K). The set of continous crossed homomorphisms from Gal(L/K) to A is denoted Z_{cts.}^{1}(Gal(L/K), A). There is a left action of A on Z_{cts.}^{1}(Gal(L/K), A) associating to a crossed homomorphism a and an element b of A, the new crossed homomorphism a^{b},

a_{g}^{b} := b^{−1} · a_{g} ·^{g}b.

Proposition 1.5. The standard description of group cohomology establishes a bijection between the set Z_{cts.}^{1}(Gal(L/K), A)/A of A-orbits and the set H^{1}(Gal(L/K), A). This bijection identifies the crossed homomorphism e with the additive identity of H^{1}(Gal(L/K), A).

Proof. It suffices to consider the case when L = L_{i} is a finite Galois extension of K. The “standard description” of group cohomology arises from the standard resolution (or unnormalized bar resolution of the trivial (left) ℤ[Gal(L_{i}/K)]-module by free ℤ[Gal(L_{i}/K)]-modules,

… → ℤ[Gal(L_{i}/K)](Gal(L_{i}/K)^{p}) ⟶^{d_{p}} ⋯ → ℤ[Gal(L_{i}/K)](Gal(L_{i}/K)×Gal(L_{i}/K))

⟶^{d_{2}} ℤ[Gal(L_{i}/K)](Gal(L_{i}/K)) ⟶^{d_{1}} ℤ[Gal(L_{i}/K)] → ℤ → 0,

d_{p}(e_{(g_{1},...,g_{p})}) = g_{1}e_{g_{2},...,g_{p}} + ∑_{i=1}^{p−1}(−1)^{p}e_{g_{1},...,g_{i−1},g_{i}·g_{i+1},g_{i+2},...,g_{p}} + (−1)^{p}e_{g_{1},...,g_{p−1}}.

Applying Hom_{ℤ[Gal(L_{i}/K)]−mod}(−, B) to this chain complex gives a cochain complex computing

H^{p}(Gal(L_{i}/K), A) = Ext_{ℤ[Gal(L_{i}/K)]−mod}^{p}(ℤ, A).

Writing out the codifferential dual to the differential above gives the “standard description” of group cohomology in terms of cocycles and coboundaries. The group of 1-cocycles is precisely the group Z^{1}(Gal(L_{i}/K), A) of crossed homomorphisms with e being the additive identity. And two crossed homomorphisms are cohomologous if and only if they are equivalent. □

1.1.3. The first group, Interpretation II.. There is a second interpretation of H^{1}(Gal(L/K), A) in terms of discrete Gal(L/K)-equivariant A-torsors.

Definition 1.6. A right A-torsor is a nonempty set E together with a right action of A which is free and transitive. A Gal(L/K)-equivariant A-torsor is a right A-torsor together with a compatible left action of Gal(L/K) on E, i.e., for every g in Gal(L/K), for every a in A and for every m in E,

g(m · a) = (^{g}m) · (^{g}a).

The Gal(L/K)-equivariant A-torsor is discrete if every element of E is stabilized by Gal(L/L_{i}) for some finite Galois subextension L_{i}/K. An isomorphism of Gal(L_{i}/K)-equivariant A-torsors is a bijection which is simultaneously left Gal(L_{i}/K)- equivariant and right A-equivariant.

Let E be a Gal(L_{i}/K)-equivariant A-torsor. For every element m of E, there is a bijection of right A-torsors

R_{m} : A → E, R_{m}(a) = m · a.

This is left Gal(L_{i}/K)-equivariant if and only if m is Gal(L_{i}/K)-invariant. The failure of m to be Gal(L_{i}/K)-invariant is measured by the map

a : Gal(L_{i}/K) → A, ∀g,^{g}m = m · a_{g}.

It is straightforward to verify a is a crossed homomorphism which is continuous if E is discrete. Moreover, replacing m by m·b^{−1} replaces the crossed homomorphism a by a^{b}. Thus the B-orbit of a is independent of the choice of m.

Proposition 1.7. Let L/K be a Galois extension and let A be a discrete Gal(L/K)-module. The rule associating to every discrete Gal(L/K)-equivariant A-torsor E the A-orbit of a in Z_{cts.}^{1}(Gal(L/K), A)/A determines a bijection between the set of isomorphism classes of discrete Gal(L/K)-equivariant A-torsors and Z_{cts.}^{1}(Gal(L/K), A)/A.

Proof. For every pair (E_{1}, m_{1}) and (E_{2}, m_{2}), there is a unique isomorphism of A-torsors

u : E_{1} → E_{2}

such that u(m_{1}) equals u(m_{2}). This isomorphism is Gal(L_{i}/K)-equivariant if and only if the crossed homomorphisms associated to (E_{1}, m_{1}) and (E_{2}, m_{2}) are equal.

Moreover, for every crossed homomorphism a, there is a pair (E, m) giving rise to a defined as follows. Define (E, m) to equal (A, e) as a right A-torsor. But endow this with a new Gal(L/K)-action

g ∗_{a} b := a_{g} ·^{g}b.

If A is discrete and a is continuous, the new action is also discrete. And the crossed homomorphism associated to (E, m) is precisely a.

This shows the rule associating to (E, m) the continuous crossed homomorphism a gives a bijection between the set of isomorphism classes of pairs (E, m) and the set Z_{cts.}^{1}(Gal(L/K), A). Since changing m to m·b^{−1} changes a to a^{b} this rule also gives a canonical bijection between the set of isomorphism classes of discrete Gal(L/K)-equivariant A-torsors and the set Z_{cts.}^{1}(Gal(L/K), A)/A. □

1.1.4. The second group. Let L_{i}/K be a finite Galois extension and let A be an Abelian Gal(L_{i}/K)-module. An extension of Gal(L_{i}/K) by A is an exact sequence of groups

0 ⟶ A ⟶ E ⟶ Gal(L_{i}/K) ⟶ 1,

i.e., a group E together with an isomorphism of A with a normal subgroup of E and an isomorphism of Gal(L_{i}/K) with the quotient group E/A such that the induced conjugation action of E on A factors through the given action of Gal(L_{i}/K) on A. An isomorphism of extensions is a commutative diagram of short exact sequences which is the identity on A and Gal(L_{i}/K).

Proposition 1.8. There is a canonical bijection between the set of equivalence classes of extensions of Gal(L_{i}/K) by A and H^{2}(Gal(L_{i}/K), A). The bijection associates the additive identity in H^{2}(Gal(L_{i}/K), A) to the equivalence class of the semidirect product A ⋊ Gal(L_{i}/K).

This is best checked by using the standard resolution to describe H^{2}(Gal(L_{i}/K), A) in terms of cocycles, cf. [Wei94, Theorem 6.6.3, p. 183].

1.1.5. Cohomological dimension. Let L/K be a Galois extension. Let p be a prime number and let n be a nonnegative integer.

Definition 1.9. The p-cohomological dimension of Gal(L/K), cd_{p}(L/K), is the smallest nonnegative integer n such that for every torsion, discrete, Abelian Gal(L/K)-module A, the p-primary component of H^{q}(Gal(L/K), A) is {0} for all q > n. If there is no such integer n, then cd_{p}(L/K) is defined to be ∞. The cohomological dimension of Gal(L/K), cd(L/K), is defined to be sup{cd_{p}(L/K)|p prime }. When L = K^{s}, a separable closure of K, these are denoted simply by cd_{p}(K) and cd(K).

2. Non-Abelian Galois cohomology and the long exact sequence

Let L/K be a Galois extension.

Definition 2.1. A Gal(L/K)-module is a (not necessarily Abelian) group A together with a left action of Gal(L/K) acting by group homomorphisms. The module is discrete if every element of A is stabilized by Gal(L/L_{i}) for some finite Galois subextension subextension L_{i}/K.

The category of Gal(L/K)-modules is not Abelian, thus one cannot define Galois cohomology in the same way as above. There is a theory of non-Abelian cohomology, developed in [Gir71] for instance. However, for H^{0} and H^{1} it is simpler to define H^{0}(Gal(L/K), A) and H^{1}(Gal(L/K), A) so that Propositions 1.4 and 1.5 continue to hold.

Definition 2.2. For every discrete Gal(L/K)-module A, the zeroth Galois cohomology is the subgroup

H^{0}(Gal(L/K), A) := A^{Gal(L/K)},

of elements of A left invariant under Gal(L/K). And the first Galois cohomology is the set of right orbits,

H^{1}(Gal(L/K), A) := Z^{1}(Gal(L/K), A)/A.

Let u : A → B be a homomorphism of discrete Gal(L/K)-modules. Since u is Gal(L/K)-equivariant, u maps A^{Gal(L/K)} to B^{Gal(L/K)}. And for every crossed homomorphism

a : Gal(L/K) → A,

the composition

u ∘ a : Gal(L/K) → B

is a crossed homomorphism. Moreover, u ∘ (a^{b}) equals (u ∘ a)^{u(b)}. Thus u ∘ a maps A-orbits into B-orbits. And u ∘ e_{A} equals e_{B}, so u preserves the distinguished points.

Definition 2.3. For every homomorphism u : A → B of discrete Gal(L/K)- modules, the group homomorphism

H^{0}(Gal(L/K), u) : H^{0}(Gal(L/K), A) → H^{0}(Gal(L/K), B)

and the map of pointed sets

H^{1}(Gal(L/K), u) : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K), B)

are as defined above.

These maps complete the definition of the functors,

H^{0}(Gal(L/K), −) : non-AbelianGal(L/K) − mod_{discrete} → Groups,

H^{1}(Gal(L/K), −) : non-AbelianGal(L/K) − mod_{discrete} → Pointed Sets.

Remark 2.4. (i) Note that the proof of Proposition 1.7 is still valid. Thus there is a canonical bijection between H^{1}(Gal(L/K), A) and the pointed set of equivalence classes of discrete Gal(L/K)-equivariant A-torsors. (ii) Note also that for a product Gal(L/K)-module, B = A × C, the functorialities of H^{0} and H^{1} establish and isomorphism

H^{0}(Gal(L/K), A × C) ≅ H^{0}(Gal(L/K), A) × H^{0}(Gal(L/K), C)

and a bijection of pointed sets,

H^{0}(Gal(L/K), A × C) ≅ H^{0}(Gal(L/K), A) × H^{0}(Gal(L/K), C).

2.1. The long exact sequence. Notice that H^{1}(Gal(L/K), A) typically has no natural structure of group, although it does have the distinguished element [e]. Thus it may seem meaningless to ask about a long exact sequence of non-Abelian Galois cohomology associated to a short exact sequence of groups. However the situation is not as bad as it might at first seem.

2.1.1. Step I. Twisting. The first ingredient in describing the “long exact sequence” is the notion of twisting. Let E be a Gal(L/K)-equivariant A-torsor and let F be a set with a left Gal(L/K)-action and a Gal(L/K)-equivariant right action of A, i.e.,

g(f · a) = ^{g}f ·^{g}a

for every g in Gal(L/K), for every a in A and for every f in F. There is a left Gal(L/K)-action on E × F by ^{g}(e, f) := (^{g}e,^{g}f). And there is a Gal(L/K)- equivariant right action of A on E × F by (e, f) · a := (e · a, f · a). The set of right A-orbits

EF := (E × F)/A

has a well-defined left action of Gal(L/K) such that [^{g}(e, f)] equals ^{g}[(e, f)].

Definition 2.5. The Gal(L/K)-set _{E}F is the twist of F by E. If E and F are both discrete, _{E}F is also discrete.

Twisting is functorial in both E and F. Since pairs (E, m) are unique up to unique isomorphism, for every element a in Z_{cts.}^{1}(Gal(L/F), A), the twist _{a}F is also well-defined up to unique isomorphism. But since two representatives a and a′ of the same A-orbit in Z_{cts.}^{1}(Gal(L, F), A) may be conjugate under many elements of A, there is no canonical isomorphism of _{a}F and _{a′}F.

If A is non-Abelian then typically _{E}F has no natural structure of A-set. However, if the action of A on F preserves some “structure”, then the twist_{E}F typically also has this “structure”. Here are two examples of this.

Example 2.6 (Translation by F). Let A and B be Gal(L/K)-modules and let F be a Gal(L/K)-set that has commuting Gal(L/K)-equivariant right actions of both A and B. Then for every Gal(L/K)-equivariant A-torsor E, _{E}F also has a Gal(L/K)-equivariant right B-action. In particular, if F is a B-torsor, then also EF is a B-torsor. Thus, associated to every Gal(L/K)-equivariant B-torsor F with a commuting Gal(L/K)-equivariant right A-action, there is an induced set map

∗F : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K), B), [E] ↦ [_{E}F].

This maps sends the trivial A-torsor to F, and thus can be considered a “translation by F ”.

The canonical example of this comes from a homomorphism u : A → B of Gal(L/K)-modules. There is a right A-action on B via b · a := bu(a). And there is a commuting “right” B-action on B via b · β := β^{−1}b. Thus, as above, there is a set map

u_{∗} : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K), B), [E] ↦ [_{E}B].

Via Proposition 1.7, the map u_{∗} is precisely the same as the map H^{1}(Gal(L/K), u).

A second example comes from the case that A is Abelian. Let B be A. For every A-torsor F, let the B-torsor structure be precisely the same as the given A-torsor structure. The “two” actions commute because multiplication in A is commutative. And the induced map ∗F literally is translation by [F] in the Abelian group H^{1}(Gal(L/K), A).

If F is itself a Gal(L/K)-module and if A acts by group homomorphisms, then EF is also a group, i.e., a Gal(L/K)-module.

Example 2.7 (Inner twists). For instance, A acts on itself by inner automorphisms. Thus associated to every discrete Gal(L/K)-equivariant A-torsor E, there is an associated Gal(L/K)-module _{E}A,

EA := (E × A)/ ∼, (m, a) ∼ (mγ, γ^{−1}aγ).

To make this more explicit, observe there is a well-defined composition,

∗ : (E × A) × (E × A) → E × A,
(m_{1}, a_{1}) ∗ (m_{2}, a_{2}) = (m_{1} · (a_{1}b), b^{−1}a_{1}b_{1}a_{2}), m_{2} = m_{1} · b.

Observe that

(m_{1} · α, α^{−1}a_{1}α) ∗ (m_{2} · β, β^{−1}a_{2}β) = ((m_{1} · (a_{1}b)) · β, β^{−1}(b^{−1}a_{1}ba_{2})β).

Thus the composition gives a well-defined composition on_{E}A. The identity element is the class of E × {e_{A}}. And the inverse of the class (m, a) is (m · a^{−1}, a). It is important to note that the composition above gives a well-defined left action of _{E}A on E = E × {e_{A}} by

[(m_{1}, a_{1})] ∗ m_{2} := m_{1}a_{1}b, m_{2} = m_{1}b.

Since _{E}A is a group, we can change this into a “right” action by

m_{2} ∗ [(m_{1}, a_{1})] := [(m_{1}a_{1}^{−1}, a_{1}^{−1})] ∗ m_{2} = m_{1}a^{−1}b.

With respect to this action, E is both a Gal(L/K)-equivariant A-torsor and a Gal(L/K)-equivariant _{E}A-torsor. Thus, by the mechanism in Example 2.6, there are well-defined set maps

∗E : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K),_{E}A),
∗E : H^{1}(Gal(L/K),_{E}A) → H^{1}(Gal(L/K), A).

Each of these maps sends the trivial torsor to E. Observe that if A is Abelian, then _{E}A equals A and the map ∗E agrees with the map from Example 2.6, i.e., it is simply translation by [E] in the Abelian group H^{1}(Gal(L/K), A).

Proposition 2.8. The maps

∗E : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K),_{E}A),
∗E : H^{1}(Gal(L/K),_{E}A) → H^{1}(Gal(L/K), A).

are both bijections.

Proof. Let F be a Gal(L/K)-equivariant A-torsor. Denote by I the set

I = Isom_{A}(E, F) = {ϕ : E → F |∀e ∈ E, ∀a ∈ A, ϕ(e · a) = ϕ(e) · a}.

Make this into a Gal(L/K)-set by defining,

(^{g}ϕ)(e) := ^{g}(ϕ(^{g^{−1}}e)), i.e., (^{g}ϕ)(^{g}e) = ^{g}(ϕ(e)).

Make this into a right _{E}A-set by defining

(ϕ · α)(e) := ϕ(e · α^{−1}), i.e., (ϕ · α)(e · α) = ϕ(e).

Putting these together,

(^{g}(ϕ · α))(^{g}(e · α)) = ϕ(e) = (^{g}ϕ ·^{g}α)(^{g}e ·^{g}α).

Since ^{g}(e · α) equals ^{g}e·^{g}α, it follows that ^{g}(ϕ · α) equals ^{g}ϕ·^{g}α, i.e., the _{E}A-action is Gal(L/K)-equivariant.

On the underlying level of groups and group-sets ignoring the Gal(L/K)- actions, E equals A as A-sets, _{E}A equals A as groups, and F equals A as A-sets. By considering this case, it is clear that the right _{E}A-action on I makes I into an EA-torsor. Therefore I is a Gal(L/K)-equivariant _{E}A-torsor.

There is a canonical map

c : I × E → F, (ϕ, e) ↦ ϕ(e).

This map is _{E}A-invariant for the diagonal right action of _{E}A on I × E. Thus it factors through a map

c′ : _{I}E → F.

Since c is Gal(L/K)-equivariant and A-equivariant for the right A-action on E, c′ is a morphism of Gal(L/K)-equivariant A-torsors. Thus it is an isomorphism. So ∗E sends I to F.

Conversely, for every Gal(L/K)-equivariant _{E}A-torsor J and for every morphism

d′ : _{J}E → F

of Gal(L/K)-equivariant A-torsors, the composition

d : J × E ↠ _{J}E ⟶^{d′} F

is A-equivariant for the right A-action on E, and thus induces a set map

d′′ : J → I.

Since d is Gal(L/K)-equivariant and _{E}A-equivariant, so is d′′. Thus d′′ is a map of Gal(L/K)-equivariant _{E}A-torsors, which is automatically an isomorphism. Therefore, up to isomorphism, I is the unique Gal(L/K)-equivariant _{E}A-torsor such that IE is isomorphic to F, i.e., the map

∗E : H^{1}(Gal(L/K),_{E}A) → H^{1}(Gal(L/K), A)

is bijective. The argument that the other map ∗E is bijective is similar. □

Let u : A → B be a homomorphism of Gal(L/K)-modules and let E be a discrete Gal(L/K)-equivariant A-torsor. There is a well-defined homomorphism of Gal(L/K)-modules,

Eu : _{E}A → _{u_{∗}E}B

associated to the map

(E × A) → (E × B), (m, a) ↦ (m, b).

The proof of the following is a straightforward exercise for the reader.

Lemma 2.9. With respect to this map, both the diagram

H^{1}(Gal(L/K), A) ⟶^{u_{∗}} H^{1}(Gal(L/K), B)
∗E↓  ↓^{∗u_{∗}E}
H^{1}(Gal(L/K),_{E}A) ⟶^{_{E}u_{∗}} H^{1}(Gal(L/K),_{u_{∗}E}B)

(1)

and the diagram

H^{1}(Gal(L/K),_{E}A) ⟶^{_{E}u_{∗}} H^{1}(Gal(L/K),_{u_{∗}E}B)
∗E↓  ↓^{∗u_{∗}E}
H^{1}(Gal(L/K), A) ⟶^{u_{∗}} H^{1}(Gal(L/K), B)

(2)

commute.

Example 2.10. Next, consider an Abelian extension of discrete Gal(L/K)- modules,

0 ⟶ A′ ⟶ A ⟶ A′′ ⟶ 1,

i.e., A′ is Abelian (although neither A nor A′′ need be Abelian). The conjugation action of A on the normal subgroup A′ determines a well-defined action of A′′ on A′. Thus, associated to every Gal(L/K)-equivariant A′′-torsor E′′ there is a discrete Abelian Gal(L/K)-module _{E′′}A′.

2.1.2. Step II. The first terms. This is enough to begin describing the long exact sequence. Let u : A → B be an injective homomorphism of discrete Gal(L/K)- modules. Denote by C = B/A the set of right A-cosets of B with its natural discrete structure of Gal(L/K)-set. Denote by v : B → C the natural map, which is Gal(L/K)-equivariant. The homomorphism u determines an injective homomorphism

H^{0}(Gal(L/K), u) : H^{0}(Gal(L/K), A) → H^{0}(Gal(L/K), B),

and in particular an action of H^{0}(Gal(L/K), A) on H^{0}(Gal(L/K), B). Because v is Gal(L/K)-equivariant, it also determines a map

v_{∗} : H^{0}(Gal(L/K), B) → C^{Gal(L/K)}.

Clearly the fibers of v_{∗} are precisely the H^{0}(Gal(L/K), A)-orbits. In this sense, the sequence of sets

1 ⟶ H^{0}(Gal(L/K), A) ⟶ H^{0}(Gal(L/K), B) ⟶ C^{Gal(L/K)}

is an exact sequence of sets.

2.1.3. Step III. The connecting map. The Gal(L/K)-invariant elements of C are the same as the A-orbits E of B which are fixed as a set (but not necessarily pointwise) by the action of Gal(L/K). Such a Gal(L/K)-invariant A-orbit E need not contain a Gal(L/K)-invariant element, i.e., v_{∗} need not be surjective. For every Gal(L/K)-invariant A-orbit E, the restrictions to E of the Gal(L/K)-action and the A-action make E into a Gal(L/K)-invariant A-torsor. Forming the equivalence class of this torsor gives a well-defined set map

δ : C^{Gal(L/K)} → H^{1}(Gal(L/K), A).

Observe that for every Gal(L/K)-invariant A-orbit E and every Gal(L/K)-invariant element b of B, left multiplication by b defines an isomorphism of the Gal(L/K)- equivariant A-torsor E and the Gal(L/K)-equivariant A-torsor b · E. Thus the map δ is constan on each H^{0}(Gal(L/K), B)-orbits in C^{Gal(L/K)}. Conversely, if E_{1} and E_{2} are Gal(L/K)-invariant A-orbits in B and if v : E_{1} → E_{2} is an isomorphism of Gal(L/K)-invariant A-torsors, then there exists a unique element b of B such that v(m) = b · m for every m in E_{1}. Since v is Gal(L/K)-equivariant, b is Gal(L/K)-invariant. Therefore δ(E_{1}) equals δ(E_{2}) if and only if E_{1} and E_{2} differ by left multiplication by an element of H^{0}(Gal(L/K), B). In particular, the fiber of δ over [A] is precisely the image of H^{0}(Gal(L/K), B).

In all of these senses, the set map

1 → H^{0}(Gal(L/K), A) → H^{0}(Gal(L/K), B) → C^{Gal(L/K)} ⟶^{δ} H^{1}(Gal(L/K), A)

is “exact”. The first map is injective. The fibers of the second map are precisely the orbits for the action of H^{0}(Gal(L/K), A) on H^{0}(Gal(L/K), B). And the fibers of the third map are precisely the orbits for the action of H^{0}(Gal(L/K), B) on (B/A)^{Gal(L/K)}.

2.1.4. Step IV. The map of torsors. Let E be a Gal(L/K)-equivariant A-torsor and let u_{∗}E be the associated Gal(L/K)-equivariant B-torsor as constructed in Example 2.6, u_{∗}E = E × B/A. Taking inverse images, a Gal(L/K)-invariant element of u_{∗}E is the same as a Gal(L/K)-invariant A-orbit F of E × B. The projection pr_{E} : F → E is an isomorphism of Gal(L/K)-invariant A-torsors. Inverting this, F is the graph F = Γ_{v} of a unique map v : E → B which is simultaneously Gal(L/K)-equivariant and A-equivariant. But then E is isomorphic to v(E), which is a Gal(L/K)-equivariant left A-orbit in B. Thus u_{∗}E is isomorphic to the distinguished element of H^{1}(Gal(L/K), B) if and only if E is isomorphic to an A-torsor in the image of δ. In summary, this gives the following.

Proposition 2.11. Associated to an injective homomorphism of Gal(L/K)- modules, u : A → B, and denoting by v : B → C the quotient map of Gal(L/K)- sets, B → B/A, the sequence of set maps

1 → H^{0}(Gal(L/K), A) ⟶^{u_{∗}} H^{0}(Gal(L/K), B) ⟶^{v_{∗}} C^{Gal(L/K)}

⟶^{δ} H^{1}(Gal(L/K), A) ⟶^{u_{∗}} H^{1}(Gal(L/K), B)

is “exact” in the following sense: the first map u_{∗} is injective, the fibers of v_{∗} equal the orbits of Image(u_{∗}), the fibers of δ equal the orbits of H^{0}(Gal(L/K), B), and the fiber of u_{∗} over the distinguished element equals the image of δ.

2.1.5. Step V. Fibers of u_{∗}. Using the twists from Example 2.7, we can push this a bit further. Let E be a Gal(L/K)-equivariant A-torsor and let

Eu : _{E}A → _{u_{∗}E}B

be the associated morphism of inner twist Gal(L/K)-modules as in Example 2.7. This gives rise to a sequence of sets,

1 → H^{0}(Gal(L/K),_{E}A) ⟶^{_{E}u_{∗}} H^{0}(Gal(L/K),_{u_{∗}E}B) ⟶^{_{E}v_{∗}} _{E}C^{Gal(L/K)}

⟶^{_{E}δ} H^{1}(Gal(L/K),_{E}A) ⟶^{_{E}u_{∗}} H^{1}(Gal(L/K),_{u_{∗}E}B)

By the diagram in Equation 2, the map ∗E defines a map from the fiber of Eu_{∗} containing the trivial _{E}A-torsor to the fiber of u_{∗} : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K), B) containing [E].

Proposition 2.12. The fiber of

u_{∗} : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K), B)

containing [E] equals the image of the map

∗E ∘_{E}δ : (_{E}C)^{Gal(L/K)} → H^{1}(Gal(L/K), A).

Proof. This follows from Lemma 2.9, Proposition 2.8 and Proposition 2.11 applied to _{E}u as discussed above. □

2.1.6. Step VI. The image of u_{∗}. Let F be a Gal(L/K)-equivariant B-torsor.

Proposition 2.13. There is Gal(L/K)-equivariant A-torsor E such that [F] equals u_{∗}[E] if and only if F/u(A) has a Gal(L/K)-invariant element.

Proof. This is straightforward. Certainly _{E}B = (E ×B)/ ∼ has a Gal(L/K)- invariant u(A)-orbit, namely the image of E × {e_{B}}. So if [F] equals u_{∗}[E], then F/u(A) has Gal(L/K)-invariant element.

Conversely, if F/u(A) has a Gal(L/K)-invariant element, the inverse image in F is a Gal(L/K)-invariant u(A)-orbit E. The restrictions of the Gal(L/K)-structure and the A-structure on F to E make E into a Gal(L/K)-equivariant A-torsor. And the inclusion E ↪ F induces a morphism _{E}B → F of Gal(L/K)-equivariant B-torsors. Thus [F] equals u_{∗}[E]. □

Observe that G/u(A) is the same as the twist _{G}C. In the sense that the twist GC should be considered “trivial” if it has a Gal(L/K)-invariant element, this extends the long exact sequence of sets one more place to the right.

2.1.7. Step VII. Quotient by a normal subgroup. If A is a normal subgroup, then C is naturally a discrete Gal(L/K)-module. In this case, for every Gal(L/K)- equivariant A-torsor E, the quotient _{E}C = _{u_{∗}E}B/_{E}A is canonically isomorphic as a Gal(L/K)-set to C. In particular, the rule associating to every element E the set map

∗E ∘_{E}δ : H^{0}(Gal(L/K), C) → H^{1}(Gal(L/K), A)

is an action of H^{0}(Gal(L/K), C) on H^{1}(Gal(L/K), A). The previous step shows that the fibers of

u_{∗} : H^{1}(Gal(L/K), A) → H^{1}(Gal(L/K), B)

are precisely the orbits of H^{0}(Gal(L/K), C). Finally, for every B-torsor G, the twist_{G}C from the previous paragraph still has a left C-action, and thus is a discrete Gal(L/K)-equivariant C-torsor. As above, it is a trivial C-torsor if and only if it has a Gal(L/K)-invariant element if and only if G is in the image of u_{∗}. In summary, this gives the following.

Proposition 2.14. Associated to an injective homomorphism of Gal(L/K)- modules, u : A → B whose image is normal, and denoting by v : B → C the quotient map of Gal(L/K)-modules, B → B/A, the sequence of set maps

1 → H^{0}(Gal(L/K), A) ⟶^{u_{∗}} H^{0}(Gal(L/K), B) ⟶^{v_{∗}} H^{0}(Gal(L/K), C)

⟶^{δ} H^{1}(Gal(L/K), A) ⟶^{u_{∗}} H^{1}(Gal(L/K), B) ⟶^{v_{∗}} H^{1}(Gal(L/K), C)

is “exact” in the following sense: the first map u_{∗} is injective with normal image, the fibers of v_{∗} equal the orbits of Image(u_{∗}), the fibers of δ equal the orbits of H^{0}(Gal(L/K), B), the fibers of the second map u_{∗} equal the orbits of H^{0}(Gal(L/K), C), and the image of the second map u_{∗} equals the fiber of the second map v_{∗} containing the neutral element.

2.1.8. Step VIII. Quotient by an Abelian normal subgroup. The next step is when A is both normal and Abelian. Then the conjugation action of B on A descends to a well-defined action of C on A. Given c in C and a in A, we will denote this action by cac^{−1} as a reminder that the action is bab^{−1} for any representative b of the A-coset c in B.

Let

c : Gal(L/K) → C

be a continuous crossed homomorphism. This in turn determines a new discrete action of G on A, namely

g ∗_{c} a := c_{g} ·^{g}a · c_{g}^{−1}.

The action of Gal(L/K) on B defines a semidirect product B ⋊ Gal(L/K). And c defines an action of B ⋊ Gal(L/K) on C by

(b, g) ∗_{c} d := c_{g} ·^{g}d · v(b)^{−1}.

The stabilizer of the identity e_{C} is a subgroup denoted _{c}H. Of course _{c}H contains A × {1} as a subgroup. The induced quotient

H ⊂ B ⋊ Gal(L/K) ⟶^{π_{2}} Gal(L/K)

is surjective and has kernel A. The conjugation action of H on A factors through an action of Gal(L/K) on A. And of course this is the new action defined above. Thus, altogether _{c}H is an extension

0 ⟶ _{c}A ⟶ _{c}H ⟶ Gal(L/K) ⟶ 0.

This gives a well-defined element

Δ(c) ∈ H^{2}(Gal(L/K),_{c}A).

For an element b of B, conjugation by b defines an automorphism of A intertwining the actions ∗_{c} and ∗_{c^{v(b)}}. Similarly conjugation by (b, 1) ∈ B ⋊ Gal(L/K) gives an isomorphism of _{c}H and _{c^{v(b)}}H which gives a morphism of extensions

0 ⟶ _{c}A ⟶ _{c}H ⟶ Gal(L/K) ⟶ 0
≅↓  ↓≅  ↓^{=}
0 ⟶ _{c^{v(b)}}A ⟶ _{c^{v(b)}}H ⟶ Gal(L/K) ⟶ 0.

In this sense, Δ(c) is independent of the choice of c.

Finally, every splitting of the extension comes from a lift of c to a crossed homomorphism Gal(L/K) → B. This proves the following.

Proposition 2.15. For every element c in Z_{cts.}^{1}(Gal(L/K), C), Δ(c) equals 0 in H^{2}(Gal(L/K), A) if and only if [c] is in the image of v_{∗}.

2.1.9. Step IX. Central extensions. The final step is when A is a central subgroup of B. Then ∗_{c} is the usual action of Gal(L/K) on A. Therefore Δ(c) is an element of H^{2}(Gal(L/K), A). Similarly, conjugation by b on A is trivial, and thus conjugation by (b, 1) defines an isomorphism of extensions of _{c}H and _{c^{v(b)}}H. Therefore Δ(c) is independent of the representative of [c], i.e., there is a well-defined set map

Δ : H^{1}(Gal(L/K), C) → H^{2}(Gal(L/K), A).

Finally, twisting A, B and C by an element c again gives a well-defined map

cΔ : H^{1}(Gal(L/K),_{c}C) → H^{1}(Gal(L/K), A).

For the same reason as in Proposition 2.12,

∗c : H^{1}(Gal(L/K),_{c}C) → H^{1}(Gal(L/K), C)

defines a bijection between the fiber of Δ containing [c] and the fiber of _{c}Δ over 0. Therefore

∗c ∘_{c}v_{∗} : H^{1}(Gal(L/K),_{c}B) → H^{1}(Gal(L/K), C)

maps surjectively onto the fiber of Δ containing [c]. In summary, this gives the following.

Proposition 2.16. Associated to a central extension of Gal(L/K)-modules,

0 ⟶ A ⟶^{u} B ⟶^{v} C ⟶ 1

the sequence of set maps

0 → H^{0}(Gal(L/K), A) → H^{0}(Gal(L/K), B) → H^{0}(Gal(L/K), C) ⟶^{δ}

H^{1}(Gal(L/K), A) ⟶^{u_{∗}} H^{1}(Gal(L/K), B) ⟶^{v_{∗}} H^{1}(Gal(L/K), C) ⟶^{Δ} H^{2}(Gal(L/K), A)

is exact in the following sense: the first map u_{∗} is injective with central image, the fibers of v_{∗} equal the orbits of Image(u_{∗}), the fibers of δ equal the orbits of H^{0}(Gal(L/K), B), the fibers of the second map u_{∗} equal the orbits of H^{0}(Gal(L/K), C), the fibers of the second map v_{∗} equal the orbits of the Abelian group H^{1}(Gal(L/K), A), and the fiber of Δ containing the element [c] equals the image of the map

∗c ∘_{c}v_{∗} : H^{1}(Gal(L/K),_{c}B) → H^{1}(Gal(L/K), C).

Remark 2.17. The long exact sequence is functorial in the short exact sequence. To be precise, given a commutative diagram of homomorphisms of Gal(L/K)- modules

A ⟶^{u} B
f↓  ↓^{g}
A′ ⟶^{u′} B′

with u and u′ injective, the diagram

C^{Gal(L/K)} ⟶^{δ_{u}} H^{1}(Gal(L/K), A)
g_{∗}↓  ↓^{f_{∗}}
(C′)^{Gal(L/K)} ⟶^{δ_{v}} H^{1}(Gal(L/K), A′)

is commutative, where C = B/A and C′ = B′/A′. Moreover, if A, resp. A′, is a central subgroup of B, resp. of B′, then also the diagram

H^{1}(Gal(L/K), C) ⟶^{Δ_{u}} H^{2}(Gal(L/K), A)
g_{∗}↓  ↓^{f_{∗}}
H^{1}(Gal(L/K), C′) ⟶^{Δ_{v}} H^{2}(Gal(L/K), A′)

is commutative. The proof of each of these assertions is a good exercise for the reader.

3. Galois cohomology of smooth group schemes

Most of the flavor of Galois cohomology comes from the specific Galois modules one studies. For instance, one could consider the action of Gal(K^{s}/K) on the étale cohomology groups of an algebraic variety defined over K. Or, when K is a number field, one could consider the action of Gal(L/K) on the ideal class group of the ring of integers in L. There is another source of Galois modules which are somewhat simpler than the previous 2 examples, but which still play an important role.

Let K be a field, let G be a quasi-compact, smooth group scheme over K, and let L/K be a Galois extension. Then the set G(L) of L-points of G is a group with a natural action of Gal(L/K) by group homomorphisms. Since G is locally finitely presented over K, G(L) is a discrete Gal(L/K)-module. In fact, this is even better than a usual Galois module. For every extension field K′/K and for every Galois extension L′/K′, the set of L′-points G(L′) is a discrete Gal(L′/K′)-module. The Galois cohomologies of these modules satisfy all the functorialities one would expect. It is a good exercise to formulate and prove these for oneself.

3.1. Galois descent of torsors. It is straightforward that H^{0}(Gal(L/K), G(L)) equals G(K). There is also an alternative description of H^{1}(Gal(L/K), G(L)) whose proof uses descent.

Definition 3.1. A right G-torsor over K is a nonempty, locally finitely presented K-scheme T together with a right action of G on T,

μ : T ×_{K} G → T

in the category of K-schemes such that the induced morphism

(pr_{T} , μ) : T ×_{K} G → T ×_{K} T

is an isomorphism. A morphism of G-torsors is a morphism of K-schemes which is equivariant for the action of G (such a morphism is automatically an isomorphism).

Let T be a G-torsor over K. There is a bijective correspondence between T (K) and the set of morphisms of G-torsors from the trivial G-torsor G to T via the rule associating to t in T (K) the morphism

μ_{t} : G → T, g ↦ μ(t, g).

In particular, T is trivial if and only if T (K) is nonempty. By the same token, for every field extension L/K, the base change T ⊗_{K} L is trivial as a G ⊗_{K} L-torsor if and only if T (L) is nonempty.

Let L/K be a Galois extension and let T be a G-torsor such that T (L) is nonempty. Then the set T (L) is naturally a Gal(L/K)-equivariant G(L)-torsor. Moreover, to every morphism

ϕ : T → T′

of G-torsors over K, the induced set map

ϕ(L) : T (L) → T′(L)

is a morphism of Gal(L/K)-equivariant G(L)-torsors. This defines a set map

p^{∗} : Hom_{G−equiv.}(T, T′) → Hom_{Gal(L/K)−G(L)−equiv.}(T (L), T′(L)), ϕ ↦ ϕ(L)

from the set of morphisms of G-torsors to the set of morphisms of Gal(L/K)- equivariant G(L)-torsors.

Proposition 3.2 (Galois descent). Let L/K be a finite Galois extension and let G be a quasi-compact, smooth group scheme over K.

(0) Every G-torsor T over K is a quasi-projective, smooth K-scheme.

(i) For every pair T, T′ of G-torsors over K such that T (L) and T′(L) are nonempty, the set map p^{∗} defined above is a bijection.

(ii) Every Gal(L/K)-equivariant G(L)-torsor is isomorphic to one associated to a G-torsor over K.

Proof. This follows by the technique of descent, cf. [Wei56, Section I], [Gro62, no. 190], [BLR90, §6.5]. □

When L/K is a Galois extension which is not necessarily finite, Proposition 3.2 still holds so long as we restrict attention to discrete torsors. In particular, the theorem establishes a bijection between H^{1}(Gal(K^{s}/K), G(L)) and the set of isomorphism classes of G-torsors.

3.1.1. Weil restriction of scalars. Let L/K be a finite extension of fields (not necessarily separable). Let X be an L-scheme.

Definition 3.3. A Weil restriction of X with respect to L/K is a pair (R_{L/K}(X), f) of a K-scheme R_{L/K}(X) together with a morphism of L-schemes

f : R_{L/K}(X) ⊗_{K} L → X

which is universal among all such pairs.

Proposition 3.4 (Weil Restriction). For every quasi-projective L-scheme X there exists a Weil restriction of X with respect to L/K. Moreover, R_{L/K}(X) is a quasi-projective K-scheme. And if X is L-smooth, then R_{L/K}(X) is K-smooth.

Proof. This follows, for instance, from [Gro62, Part IV.4.c, p. 221-19]. See also [BLR90, Theorem 4, p. 194]. The smoothness assertion is an easy application of the infinitesimal lifting criterion for smoothness, cf. [Gro67, Théorème 17.5.1] or [Har77, Exercise II.8.6]. □

For every quasi-compact, smooth group scheme G_{L} over L, the Weil restriction R_{L/K}(G_{L}) has a natural structure of a group scheme over K. For every G_{L}-torsor T_{L} over L, the Weil restriction R_{L/K}(T_{L}) has a natural structure of a R_{L/K}(G_{L})- torsor.

Also, for every K-scheme Y, denoting by Y_{L} the base change Y ⊗_{K} L, the identity morphism

Y ⊗_{K} L → Y_{L}

determines a K-morphism

i_{Y} : Y → R_{L/K}(Y_{L}).

For every open affine U = Spec R of Y, for every K-algebra A and for every pair of morphisms of K-algebras,

ϕ, ϕ′ : R → A,

if the associated A ⊗_{K} L-points,

ϕ ⊗ Id_{L}, ϕ′ ⊗ Id_{L} : R → A ⊗_{K} L

are equal, then ϕ and ϕ′ are equal; indeed, the map of K-vector spaces

Hom_{K−v. space}(R, A) → Hom_{K−v. space}(R, A ⊗_{K} L)

is injective since A → A ⊗_{K} L is injective. Thus, for a quasi-projective K-scheme Y, it follows that i_{Y} is an unramified monomorphism.

Proposition 3.5. Let L/K be a finite Galois extension.

(i) For every quasi-compact, smooth group scheme G over K,

i_{G} : G → R_{L/K}(G_{L})

is both a closed immersion and a homomorphism of quasi-compact, smooth group schemes over K.

(ii) The connecting map in Galois cohomology,

δ : (R_{L/K}(G_{L})/G)(K) → H^{1}(Gal(L/K), G(L))

is surjective, i.e., every G-torsor over K having an L-point is isomorphic to a G-orbit in R_{L/K}(G_{L}).

Proof. (i). First of all, since G is a quasi-compact, smooth group scheme over K, it is quasi-projective, cf. [BLR90, Theorem 1, p. 153] for instance. Thus G_{L} is also a quasi-projective, smooth group scheme over L. Thus, by Proposition 3.4, R_{L/K}(G_{L}) is a quasi-projective, smooth group scheme over K. For every K-scheme T, the pullback map G(T) → G(T ⊗_{K} L) is a group homomorphism. Thus also i_{G}(T) is a group homomorphism, i.e., i_{G} is a homomorphism of group schemes. And by the usual argument, the image of i_{G} is a closed subgroup scheme of R_{L/K}(G_{L}): the closure of i_{G}(G) is a group and i_{G}(G) is a dense, equivariant open in this group, and therefore the entire group. A homomorphism of group schemes which is an unramified monomorphism is automatically a closed immersion; the closure of the image is a group scheme which contains i_{G}(G) is an equivariant, dense open, hence as all of the closure.

(ii). Let T be a G-torsor over K. If T (L) is nonempty, then there is an isomorphism of G_{L} and T_{L} as G_{L}-torsors over L. Thus there is an isomorphism

u : R_{L/K}(T_{L}) ⟶^{≅} R_{L/K}(G_{L})

as R_{L/K}(G_{L})-torsors over K. The morphism

i_{G} : G → R_{L/K}(G_{L})

is a homomorphism of group schemes over K. And the composition

u ∘ i_{T} : T → R_{L/K}(G_{L})

embeds T as an i_{G}(G)-orbit in R_{L/K}(G_{L}). □

Part (ii) can also be proved by writing out explicitly the map

H^{1}(Gal(L/K), G(L)) ⟶^{i_{G,∗}} H^{1}(Gal(L/K), R_{L/K}(G_{L})(L))

and verifying it is constant.

3.2. Examples.

3.2.1. The multiplicative group. One particularly important case is when G = 𝔾_{m,K}, the multiplicative group scheme Spec K[t, t^{−1}].

Theorem 3.6 (Hilbert’s Theorem 90). For the multiplicative group, H^{0}(Gal(K^{s}/K), 𝔾_{m,K}(K^{s})) equals K^{∗} and H^{1}(Gal(K^{s}/K), 𝔾_{m,K}(K^{s})) is {[𝔾_{m,K}(L)]}.

Proof. The first assertion is obvious. For the second assertion, it suffices to verify H^{1}(Gal(L/K), 𝔾_{m,K}(L)) is trivial for every finite Galois extension L/K, i.e., every 𝔾_{m,K}-torsor having an L-point also has a K-point. By Proposition 3.5, every 𝔾_{m,K}-torsor having an L-point is a 𝔾_{m,K}-orbit in R_{L/K}(𝔾_{m,L}). Of course 𝔾_{m,K} is an open subscheme of the affine line 𝔸_{K}^{1}, and the immersion is equivariant. Thus R_{L/K}(𝔾_{m,L}) is an equivariant open subscheme of R_{L/K}(𝔸_{L}^{1}). Up to choosing a basis {b_{1}, …, b_{n}} for L as a K-vector space, R_{L/K}(𝔸_{L}^{1}) is isomorphic to 𝔸_{K}^{n} as a K-scheme. Moreover, the action of 𝔾_{m,K} on R_{L/K}(𝔸_{L}^{1}) induced by the homomorphism

i_{𝔾_{m,K}} : 𝔾_{m,K} → R_{L/K}(𝔾_{m,L})

equals the standard action of 𝔾_{m,K} on 𝔸_{K}^{n} via this isomorphism. So the 𝔾_{m,K}- orbits in R_{L/K}(𝔸_{K}^{1}) are the same as the 𝔾_{m,K}-orbits in 𝔸_{K}^{n}, i.e., the set of nonzero elements in 1-dimensional K-linear subspaces of 𝔸_{K}^{n}. Therefore Hilbert’s Theorem 90 reduces to the statement that every 1-dimensional K-linear subspace of 𝔸_{K}^{n} contains a nonzero vector, which is obvious. □

Of course there is another proof which forms a Poincaré series out of a 1-cocycle and then uses linear independence of automorphisms to conclude this Poincaré series is nonzero. In fact the proof above uses the full machinery of descent, parts of which are proved by using arguments very similar to the Poincaré series argument.

Because of Hilbert’s Theorem 90, the first nontrivial cohomology group is H^{2}(Gal(K^{s}/K), 𝔾_{m,K}(K^{s})).

Definition 3.7. For every Galois extension L/K, the cohomological Brauer group of L/K, denoted Br′(L/K), is H^{2}(Gal(L/K), 𝔾_{m,K}(L)). When L is a separable closure K^{s} of K, this is called the cohomological Brauer group of K, Br′(K).

3.2.2. Groups of roots of unity. Let n be a positive integer not divisible by char(K). Then the group scheme μ_{n,K} := Spec K[t, t^{−1}]/⟨t^{n} − 1⟩ is a smooth group scheme over K. There is a short exact sequence of smooth, commutative group schemes over K, the Kummer sequence,

1 ⟶ μ_{n,K} ⟶ 𝔾_{m,K} ⟶^{(−)^{n}} 𝔾_{m,K} ⟶ 1.

Proposition 3.8. The Kummer sequence induces isomorphisms of Abelian groups,

H^{1}(Gal(K^{s}/K), μ_{n,K}(K^{s})) = K^{∗}/(K^{∗})^{n}

and

H^{2}(Gal(K^{2}/K), μ_{n,K}(K^{s})) = Br′(K)[n].

Proof. This follows immediately from the long exact sequence of (Abelian) Galois cohomology and Theorem 3.6. □

3.2.3. Additive and unipotent groups. Another important case is when G = 𝔾_{a,K} = Spec K[t], the additive group.

Proposition 3.9. For the additive group 𝔾_{a,K}, H^{0}(Gal(K^{s}/K), 𝔾_{a,K}(K^{s})) equals K and for every q > 0, H^{q}(Gal(K^{s}/K), 𝔾_{a,K}(K^{s})) equals {0}.

Proof. For every finite Galois extension L/K with Galois group Γ, the normal basis theorem gives an isomorphism between L and K[Γ] as K[Γ]-modules. Since K[Γ] is a free module over itself, it has no higher Galois cohomology. □

If char(K) = p, then there is a short exact sequence of smooth, commutative group schemes, the Artin-Schreier sequence,

0 ⟶ ℤ/pℤ ⟶ 𝔾_{a,K} ⟶^{F} 𝔾_{a,K} ⟶ 0

where F (a) = a^{p} − a.

Corollary 3.10. If char(K) = p, then H^{0}(Gal(K^{s}/K), ℤ/pℤ) equals ℤ/pℤ, H^{1}(Gal(K^{s}/K), ℤ/pℤ) equals K/F (K), and for every q > 1, H^{q}(Gal(K^{s}/K), ℤ/pℤ) equals {0}.

Corollary 3.11. If a quasi-compact, smooth group scheme U over K has a composition series by normal, closed subgroup schemes whose subquotients are each isomorphic to 𝔾_{a,K}, then H^{1}(Gal(K^{s}/K), U(K^{s})) equals {∗}.

The first corollary follows from Proposition 3.9 and the long exact sequence of Abelian Galois cohomology. The second corollary follows by inductively applying Proposition 2.14 to the terms in the composition series, using Proposition 3.9 for the base case of the induction.

Definition 3.12. A connected, quasi-compact, smooth group scheme U over K is unipotent if for some integer N, U is isomorphic to a closed subgroup scheme of GL_{N,K} whose K^{s}-points are all unipotent (i.e., 1 is the only eigenvalue).

Remark 3.13. If K is a perfect field, then every connected, unipotent group over K has a composition series whose subquotients are each isomorphic to 𝔾_{a,K}. Thus H^{1}(Gal(K^{s}/K), U(K^{s})) equals {∗} for every connected, unipotent group U over a perfect field K.

3.2.4. Matrix groups. For every integer n, the affine group scheme GL_{n,K} over K, resp. SL_{n,K}, PGL_{n,K} over K, represents the functor

K − algebras → Groups, A ↦ Isom_{A−mod}(A^{⊕n}, A^{⊕n}),

resp.

K − algebras → Groups, A ↦ {ϕ ∈ Isom_{A−mod}(A^{⊕n}, A^{⊕n})|det(ϕ) = 1},

K − algebras → Groups, A ↦ Isom_{A−mod}(A^{⊕n}, A^{⊕n})/ ∼, ϕ ∼ λ · ϕ, ∀λ ∈ A^{∗}. Basically the same, for every finite dimensional K-vector space V, the group schemes GL_{V,K}, resp. SL_{V,K}, PGL_{V,K}, represents the functor

K − algebras → Groups, A ↦ Isom_{A−mod}(A ⊗_{K} V, A ⊗_{K} V),

resp.

K − algebras → Groups, A ↦ {ϕ ∈ Isom_{A−mod}(A ⊗_{K} V, A ⊗_{K} V)|det(ϕ) = 1},

K−algebras → Groups, A ↦ Isom_{A−mod}(A⊗_{K} V, A⊗_{K} V)/ ∼, ϕ ∼ λ·ϕ, ∀λ ∈ A^{∗}.

Of course up to choosing a basis for V, these are the same groups.

Theorem 3.14 (Speiser’s Theorem). For every integer n ≥ 1, every GL_{n,K}- torsor over K, resp. every SL_{n,K}-torsor over K, is trivial.

Proof. This is almost the same as the proof of Theorem 3.6. By the same argument as there, for every finite Galois extension L/K, every GL_{n,K}-torsor over K with an L-point is isomorphic as a GL_{n,K}-torsor to a free (right) GL_{n,K}-orbit in Hom_{K−v. space}(K^{⊕n}, L^{⊕n}). Every free orbit is of the form Isom_{K−v. space}(K^{⊕n}, V) where V is an n-dimensional K-linear subspace of L^{⊕n}. Choosing a basis for V, such an orbit has a K-point.

Consider the short exact sequence of group schemes

1 ⟶ SL_{n,K} ⟶ GL_{n,K} ⟶^{det} 𝔾_{m,K} ⟶ 0.

This is “split” by the homomorphism

s : 𝔾_{m,K} → GL_{n,K},

λ ↦ [ \begin{matrix}λ & 0 & … & 0 \\ 0 & 1 & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & 1 \end{matrix} ] .

Thus the connecting map in the associated long exact sequence of Galois cohomology is constant. Therefore H^{1}(Gal(K^{s}/K), SL_{n,K}(K^{s})) is a subset of H^{1}(Gal(K^{s}/K), GL_{n,K}(K^{s})), which is a singleton set by the previous paragraph. □

The long exact sequence of Galois cohomology associated to the central extension

1 ⟶ 𝔾_{m,K} ⟶^{(−)·Id_{n×n}} GL_{n,K} ⟶^{q} PGL_{n,K} ⟶ 1

gives a connecting map

Δ_{n} : H^{1}(Gal(L/K), PGL_{n,K}(L)) → H^{2}(Gal(L/K), 𝔾_{m,K}(L)) = Br′(L/K).

Theorem 3.15. (i) The fiber of Δ_{n} over the neutral element is {∗}. (ii) The image of Δ_{n} is contained in the n-torsion subgroup, Br′(L/K)[n].

Proof. (i). This follows from Proposition 2.16 and Theorem 3.14.

(ii). By Remark 2.17, associated to the following commutative diagram of central extensions of smooth, linear group schemes,

1 ⟶ 𝔾_{m,K} ⟶^{(−)·Id_{n×n}}  GL_{n,K}  ⟶^{q} PGL_{n,K} ⟶ 1
(−)^{n}↓  ↓^{(det,q)}  ↓^{=}
1 ⟶ 𝔾_{m,K} ⟶^{e_{1}} 𝔾_{m,K} × PGL_{n,K} ⟶^{π_{2}} PGL_{n,K} ⟶ 1

there is a commutative diagram

H^{1}(Gal(L/K), PGL_{n,K}(L)) ⟶^{Δ_{n}} H^{2}(Gal(L/K), 𝔾_{m,K}(L)) ^{=}↓  ↓^{(−)^{n}} H^{1}(Gal(L/K), PGL_{n,K}(L)) ⟶^{Δ_{e_{1}}} H^{2}(Gal(L/K), 𝔾_{m,K}(L)).

Since e_{1} is split, Δ_{e_{1}} is the constant map. Thus the image of Δ_{n} is in the kernel of the raising to the n^{th} power map, i.e., it is in Br′(L/K)[n]. □

For two finite dimensional K-vector spaces V and W, there is a morphism of smooth group schemes,

⊗ : GL_{V,K} × GL_{W,K} → GL_{V ⊗_{K}W,K}, (ϕ, ψ) ↦ ϕ ⊗ ψ.

This induces morphisms of smooth group schemes,

⊗ : SL_{V,K} × SL_{W,K} → SL_{V ⊗_{K}W,K},

and

⊗ : PGL_{V,K} × PGL_{W,K} → PGL_{V ⊗_{K}W,K}.

In particular, by the functoriality of Galois cohomology and by Remark 2.4, this induces a map of pointed sets,

⊗ : H^{1}(Gal(L/K), PGL_{V,K}(L))×H^{1}(Gal(L/K), PGL_{W,K}(L)) → H^{1}(Gal(L/K), PGL_{V ⊗_{K}W,K}(L)).

Up to choosing bases, this is the same as a map

⊗ : H^{1}(Gal(L/K), PGL_{l,K}(L))×H^{1}(Gal(L/K), PGL_{n,K}(L)) → H^{1}(Gal(L/K), PGL_{ln,K}(L)).

Lemma 3.16. For every PGL_{l,K}-torsor E, resp. PGL_{n,K}-torsor F, with an L-point, Δ_{ln}(E ⊗ F) equals Δ_{l}(E) + Δ_{n}(F).

Proof. By Remark 2.17, associated to the following commutative diagram of central extensions of smooth, linear group schemes,

1 ⟶ 𝔾_{m,K} × 𝔾_{m,K} ⟶ GL_{l,K} × GL_{n,K} ⟶ PGL_{l,K} × PGL_{n,K} ⟶ 1 ^{×}↓  ↓^{⊗}  ↓^{⊗} 1 ⟶  𝔾_{m,K}  ⟶  GL_{lm,K}  ⟶  PGL_{lm,K}  ⟶ 1,

there is a commutative diagram

H^{1}(Gal(L/K), PGL_{l,K}(L)) × H^{1}(Gal(L/K), PGL_{n,K}(L)) ⟶^{Δ_{l}×Δ_{n}} Br′(L/K) × Br′(L/K) ^{⊗}↓  ↓^{×} H^{1}(Gal(L/K), PGL_{ln,K}(L))  ⟶^{Δ_{ln}}  Br′(L/K).

The lemma is equivalent to the commutativity of this diagram. □

4. The Brauer group

Definition 4.1. Let L/K be a Galois extension. A central simple algebra over L/K is a finite dimensional K-algebra A (so K is contained in the center of A) such that A⊗_{K} L is isomorphic as an L-algebra to the matrix algebra Mat_{n×n,L} for some integer n. When L is a separable closure K^{s}, this is simply called a central simple algebra over K.

In particular, matrix algebras Mat_{n×n,K} are central simple algebras over K. If A is a central simple algebra over K, then so is the opposite algebra A^{opp} defined to be A but with opposite multiplication a∗b := ba. And if A and A′ are central simple algebras over K, so is A⊗_{K} A′. Thus the set of isomorphism classes of central simple algebras is an associative, commutative semigroup with identity element [K]. But it is better than this.

Definition 4.2. Two central simple algebras A and A′ are Morita equivalent if there exist positive integers n and n′ such that Mat_{n×n,K} ⊗_{K} A is isomorphic to Mat_{n′×n′,K} ⊗_{K} A′.

Proposition 4.3. Let A, A′ and B be central simple algebras over L/K. If A and A′ are Morita equivalent, so are A^{opp} and (A′)^{opp}, and so are A ⊗_{K} B and A′ ⊗_{K} B. Also, the natural map

A ⊗_{K} A^{opp} → Hom_{K−v. space}(A, A),
(a, b) ↦ (c ↦ acb)

is an isomorphism of central simple algebras. In fact the composition with the trace,

A ⊗_{K} A^{opp} → Hom_{K−v. space}(A, A) ⟶^{Trace} K

is a perfect pairing of K-vector spaces.

Proof. The assertions about Morita equivalence are straightforward. The last two statements say that a certain maps of K-vector spaces are isomorphisms. These may be checked after base change from K to K^{s}, and then A is isomorpic to a matrix algebra. The assertions are easy to verify for matrix algebras (using the bases of elementary matrices, for instance). □

Because of the proposition, the set of Morita equivalence classes is an Abelian group under [A] + [B] = [A ⊗_{K} B] and −[A] = [A^{opp}]. The identity element is 0 = [Mat_{n,K}] for any integer n ≥ 1.

Definition 4.4. Let L/K be a Galois extension. The group of Morita equivalence classes of central simple algebras over L/K is the Brauer group of L/K, denote Br(L/K). When L is a separable closure K^{s} of K, this is referred to as the Brauer group of K, Br(K).

4.1. Relation to PGL-torsors. Let n be a positive integer. The group PGL_{n,K} acts by algebra automorphisms on Mat_{n×n,K} via conjugation. Thus, for every PGL_{n,K}-torsor E over K, the twist

A_{E} := _{E}Mat_{n×n,K} = (E × Mat_{n×n,K})/PGL_{n,K}

is a central simple algebra of dimension n^{2}. For a field extension L/K, if E has an L-point, then A_{E} ⊗_{K} L is isomorphic to Mat_{n×n,L}. This defines a set map,

A_{−} : H^{1}(Gal(L/K), PGL_{n,K}(L)) → A_{n}(L/K)

where A_{n}(L/K) is the set of isomorphism classes of central simple algebras A over K such that A ⊗_{K} L ≅ Mat_{n×n,L}.

Proposition 4.5 (Skolem-Noether). The set map above is a bijection. Moreover, for PGL_{n,K}-torsors E and F, the induced map from the set of morphisms in the category of torsors to the set of morphisms in the category of central simple algebras,

A_{−} : Hom_{PGL_{n,K}} (E, F) → Hom_{K−CSA}(A_{E}, A_{F}),

is a bijection.

Proof. Let A be a central simple algebra in A_{n}(L/K). Define E to be the affine K-scheme representing the functor

K − algebras → Sets, R ↦ Isom_{R−algebra}(Mat_{n×n,R}, A ⊗_{K} R).

There is a right action of PGL_{n,K} on E by precomposing isomorphisms with automorphisms of Mat_{n×n,K}. The Skolem-Noether theorem says that this action makes I a PGL_{n,K}-torsor. In fact this can be checked after base change from K to K^{s}, so it reduces to the classical formulation of the Skolem-Noether theorem: every K-algebra endomorphism of Mat_{n×n,K^{s}} comes from PGL_{n,K}(K^{s}). The natural map

E × Mat_{n×n,K} → A

is invariant for the natural PGL_{n,K}-action and thus factors through a K-algebra homomorphism

ϕ : A_{E} → A.

Since this is a K-algebra homomorphism of central simple algebras, it is automatically an isomorphism (this can be checked after base change to K^{s} where it follows from Skolem-Noether).

For every PGL_{n,K}-torsor F and every K-algebra homomorphism

ψ : A_{F} → A,

the composition

F × Mat_{n×n,K} → A

induces a morphism of schemes from τ : F → E. It is straightforward to verify this is an isomorphism of PGL_{n,K}-torsors, and is the unique isomorphism such that ψ = ϕ ∘ A_{τ}. The second part of the proposition follows. □

4.2. The cohomological Brauer group equals the Brauer group. Because of Proposition 4.5, the map from Theorem 3.15 factors through a well-defined set map,

Δ_{n} : A_{n}(L/K) → Br′(L/K), Δ_{n}(A_{E}) := Δ_{n}(E).

Lemma 4.6. For every [A] in A_{l}(L/K) and for every [B] in A_{n}(L/K), Δ_{ln}(A⊗_{K} B) equals Δ_{l}(A) + Δ_{n}(B). In particular, if A and B are Morita equivalent, then Δ_{l}(A) equals Δ_{n}(B). Thus there is a well-defined group homomorphism,

Δ : Br(L/K) → Br′(L/K), Δ([A]) = Δ_{l}(A).

This group homomorphism is injective.

Proof. For a PGL_{l,K}-torsor E and a PGL_{n,K}-torsor F, A_{E⊗F} equals A_{E} ⊗_{K} A_{F} as central simple algebras (this is easiest to verify using cocycles). Thus Lemma 3.16 implies that Δ_{ln}(A_{E} ⊗_{K} A_{F}) equals Δ_{l}(A_{E}) + Δ_{n}(A_{F}). Together with Proposition 4.5, this implies the first assertion.

Since Δ_{n}(E) equals 0 for the trivial PGL_{n,K}-torsor, also Δ_{n}(Mat_{n×n,K}) equals 0. Thus, by the previous paragraph, Δ_{ln}(A⊗_{K} Mat_{n×n,K}) equals Δ_{l}(A). Therefore Δ_{l}(A) equals Δ_{n}(B) if A and B are Morita equivalent. Thus there is a well-defined map

Δ : Br(L/K) → Br′(L/K)

such that for every integer l ≥ 1 and every A in A_{l}(L/K), Δ([A]) equals Δ_{l}(A).

Finally, assume Δ([A]) equals Δ([B]). Then A^{opp} ⊗_{K} B and Mat_{ln×ln,K} are elements of A_{ln}(L/K) with the same image under Δ_{ln}. By Theorem 3.15(i) and Proposition 4.5, this implies that A^{opp} ⊗_{K} B is isomorphic to Mat_{ln×ln,K} as central simple algebras, i.e., [B] − [A] equals 0 in Br(L/K). Therefore [A] equals [B] in Br(L/K). So Δ is injective. □

Theorem 4.7. Let L/K be a finite Galois extension of degree n. Then

Δ_{n} : A_{n}(L/K) → Br′(L/K)

is surjective. Thus for every Galois extension L/K (not necessarily finite), the homomorphism

Δ : Br(L/K) → Br′(L/K)

is surjective, and hence bijective. In other words, the Brauer group and the cohomological Brauer group are canonically isomorphic.

Proof. Using Proposition 1.8, every element in Br′(L/K) comes from an extension

1 ⟶ L^{∗} ⟶^{u} M ⟶^{v} Gal(L/K) ⟶ 1.

The group algebra K[M] is an associative K-algebra which is a free K-vector space with basis {e_{m}|m ∈ M}. There is an action of M on L by K-algebra homomorphisms through the quotient v. This defines a multiplication on the tensor product

L[M] := L ⊗_{K} K[M], (a ⊗ e_{m}) · (b ⊗ e_{n}) := (a_{v(m)}b) ⊗ e_{mn}

for elements a, b ∈ L and m, n ∈ M. It is straightforward to verify this multiplication is K-linear and associative with 1 ⊗ 1_{N} as an identity element.

Define I to be the 2-sided ideal in L[M] defined by the generators,

I := ⟨λ ⊗ e_{1_{M}} − 1 ⊗ e_{u(λ)}|λ ∈ L^{∗}⟩.

Denote the quotient algebra by A_{M} := L[M]/I. As a (left) L-vector space, it is straightforward to check that A_{M} has dimension equal to #Gal(L/K) = n. This algebra is usually denoted as the cross-product algebra associated to the extension M.

It is a somewhat involved exercise to check that A_{M} is a central simple algebra and that Δ_{n}(A_{M}) equals [M]. This is usually done by choosing a set map splitting s : Gal(L/K) → M of v and using this to translate everything into cocycles. Since the details do not seem very illuminating, they are left to the interested reader, cf. [Ser79, Exercises 1 and 2, p.159]. □

4.3. Division algebras.

Definition 4.8. A division algebra over L/K is a central simple algebra A over L/K such that every nonzero element of A is invertible. Equivalently, it is a central simple algebra whose only right ideals are {0} and A (and the same also holds with left ideals in place of right ideals). When L is a separable closure K^{s} of K, these are referred to simply as division algebras over K.

Lemma 4.9. Let A be a division algebra over K Every finitely generated left (or right) A-module is free.

Proof. Let M be a left A-module. If M equals {0} then the assertion is vacuous, M = A^{⊕0}. Thus assume M is nonzero. Consider a surjective homomorphism of D-modules,

q : A^{⊕n} ↠ M

for which the integer n is minimal. Denote by N the kernel of q. If N is nonempty, then for some integer 1 ≤ i ≤ n, the projection

π_{i} : K → A

is nonzero. Thus π_{i}(N) is a nonzero left A-submodule of A, i.e., a left ideal in A. Since A is a division algebra, π_{i}(N) equals A. But this means that, modulo N, the i^{th} basis element of A^{⊕n} is congruent to a linear combination of the other basis elements. So the restriction of q to the free submodules generated by the other basis elements is still surjective, contradicting the minimality of n. Therefore N equals {0}, i.e., q is an isomorphism of A-modules. □

In order to prove the next result, it is useful to describe the right ideals in Mat_{n×n,K}. The proof is left as an exercise for the reader.

Lemma 4.10. For a nonzero, finite dimensional K-vector space V, the right ideals of Mat_{V,k} are in 1-to-1 correspondence with the K-linear subspaces W of V via the rule,

W ↦ Hom_{K−v. space}(V, W).

Lemma 4.11 (Schur’s Lemma). Let A be a central simple algebra over K. Assume that A is not a division algebra, i.e., A has a proper, nonzero right ideal I. Let I be a proper, nonzero right ideal which is minimal, i.e., has minimal dimension as a K-vector space. Then every nonzero left A-module homomorphism,

ϕ : I → I

is an isomorphism and the K-algebra of A-module homomorphisms,

D := End_{A−mod}(I, I)

is a division algebra.

Proof. The image ϕ(I) is a left ideal in A. Since I is minimal, if ϕ is nonzero then ϕ(I) equals I, i.e., ϕ is surjective. A surjective K-linear endomorphism of a finite dimension K-vector space is injective, thus ϕ is an isomorphism. Thus every nonzero element of D is nonzero. After base change from K to L, A ⊗_{K} L is isomorphic to Mat_{V,L} for some nonzero, finite dimensional L-vector space V. By Lemma 4.10, I ⊗_{K} L is isomorphic to Hom_{L−v. space}(V, W) for some nonzero, proper L-subspace W of V. It is straightforward to check that

End_{Mat_{V,L}}(Hom_{L−v. space}(V, W)) = Mat_{W,L}.

Thus D ⊗_{K} L is isomorphic to Mat_{W,L}, i.e., D is a central simple algebra over K. Therefore D is a division algebra. □

There is an induced left action of D on I which commutes with the right action of A on I. Thus there is an induced K-algebra homomorphism,

A → End_{D}(I).

Theorem 4.12 (Artin-Wedderburn Theorem). Every central simple algebra A over L/K is isomorphic to D ⊗_{K} Mat_{n×n,K} for a unique division algebra D over L/K.

Proof. By base change from K to L, one can check that the K-linear map

A → End_{D}(I)

above is an isomorphism. By Lemma 4.9, I is a free D-module, say I ≅ D^{⊕n}. Therefore A is isomorphic to D ⊗_{K} Mat_{n×n,K}.

Let D be a division algebra and let A be D ⊗_{K} Mat_{n×n,K}. Then, up to isomorphism, the unique (nonzero) simple A-module is M := D^{⊕n} with its natural A-action. And the natural map

D → End_{A−mod}(M)

is an isomorphism of algebras. Therefore D is uniquely determined by A. □

4.4. The period-index problem.

Definition 4.13. Let A be a central simple algebra over K. The index of A, denote index(A), is the unique integer n such that dim_{K}(D) = n^{2} where D is a division algebra Morita equivalent to A. The period or exponent of A, denoted period(A), is the order of [A] in Br(K). A splitting field for A is a separable field extension L/K such that A ⊗_{K} L is isomorphic to Mat_{m×m,L} for some integer m.

It turns out that index(A) equals both the minimal degree [L : K] of a splitting field of A as well as the greatest common divisor of the degrees [L : K] of all splitting fields.

Proposition 4.14 (R. Brauer). For every central simple algebra A over K, period(A) divides index(A), and both integers have the same prime factors (but often with different multiplicities).

Proof. See [Ser79, Exercise 3, p. 159]. □

In light of this proposition, it is natural to ask about the precise relation between the period and the index.

Problem 4.15 (The Period-Index Problem). Let K be a field. If there exists a nonnegative integer r such that index(A) divides period(A)^{r} for every central simple algebra A, the least such integer is the power of K, denoted r(K); otherwise r(K) is defined to be ∞. What is the power of K? For a given integer r, what “conditions” on K guarantee that r(K) ≤ r?

Of course this problem is ill-posed since the meaning of “condition” is not specified. In Chapter 4 we will explain some answers which hopefully give a clearer meaning to this problem.

5. The universal cover sequence

There is a generalization of the map Δ_{n} from Theorem 3.15 to other linear algebraic group schemes. The generalization uses some of the structure theory for linear algebraic group schemes. Unipotent groups were defined in Definition 3.12.

Definition 5.1. A connected, quasi-compact, smooth group scheme T is multiplicative or of multiplicative type if for some integer n ≥ 0, T ⊗_{K} K^{s} is isomorphic to 𝔾_{m,K^{s}}^{n} as a group scheme over K^{s}.

For a quasi-compact, smooth group scheme G over K, the unipotent radical of G, R_{u}(G), is the maximal connected, normal subgroup scheme of G which is unipotent. If R_{u}(G) is trivial, then G is reductive.

For a connected, smooth, linear algebraic group scheme G over K, the multiplicative quotient of G, T (G), is the maximal quotient group of G (by a closed, normal subgroup scheme) which is multiplicative. If R_{u}(G) and T (G) are trivial, then G is semisimple. If T (G) is the quotient of G by R_{u}(G), then G is solvable. The maximal connected, normal, solvable subgroup of G is the solvable radical of G, or simply the radical of G, denoted R(G).

Let G be a connected, smooth, linear algebraic group scheme over K. Of course R_{u}(G) is the unipotent radical of R(G). And R_{u}(G) is contained in the kernel of G → T (G). Thus there is an induced homomorphism of group schemes over K,

R(G)/R_{u}(G) → T (G).

This is a finite and faithfully flat morphism, i.e., an isogeny of group schemes. But it is not necessarily an isomorphism. (Notice however that T (G) is trivial if and only if R(G) equals R_{u}(G), so that G is semisimple if and only if R(G) is trivial.) Because T (G) and R(G)/R_{u}(G) are different, there are 2 different decompositions that arise naturally in the study of group schemes.

Notation 5.2. Let G be a connected, smooth, linear algebraic group scheme over K. The first associated semisimple group of G is the group

L(G) = Ker(G → T (G))/R_{u}(G).

The second associated semisimple group of G is the group

L′(G) = G/R(G).

As the definition of L′(G) is simpler, it is the one that arises most often. But L(G) is slightly better behaved with respect to universal covers and fundamental groups. Both of these definitions should be considered as decompositions of G into a unipotent group, R_{u}(G), a multiplicative group, T (G), resp. R(G)/R_{u}(G), and a semisimple group, L(G), resp., L′(G).

Definition 5.3. A connected, semisimple algebraic group L is simply connected if for every connected, semisimple algebraic group M, every isogeny

M → L

is an isomorphism. For every connected, semisimple algebraic group L, there exists a connected, simply connected, semisimple algebraic group L̃ and an isogeny

L̃ → L,

called the universal cover of L. The kernel of the universal cover, denote π_{1}L, is the fundamental group scheme of L.

There is a central extension of group schemes,

0 ⟶ π_{1}L ⟶ L̃ ⟶ L ⟶ 1

Note that π_{1}L is sometimes not smooth, e.g., π_{1}PGL_{n,K} is the non-smooth group scheme μ_{n,K} when char(K) divides n. There is a generalization of Galois cohomology, cohomology for the fppf topology, H_{fppf}^{1}(K; −). This agrees with Galois cohomology for smooth group schemes. This cohomology theory also has a long exact sequence, leading to a sequence

0 → H_{fppf}^{0}(K; π_{1}G) → H^{0}(Gal(K^{s}/K), G̃(K^{s})) → H^{0}(Gal(K^{s}/K), G(K^{s})) ⟶^{δ}

H_{fppf}^{1}(K, π_{1}G) ⟶^{u_{∗}} H^{1}(Gal(K^{s}/K), G̃(K^{s})) ⟶^{v_{∗}} H^{1}(Gal(K^{s}/K), G(K^{s})) ⟶^{Δ} H_{fppf}^{2}(K, π_{1}G).

This sequence satisfies the same conditions as in Proposition 2.16.

Serre posed two conjectures about this sequence. The first conjecture was proved by R. Steinberg.

Theorem 5.4 (Steinberg’s Theorem = Serre’s “Conjecture I”). [Ste65] If K has cohomological dimension ≤ 1, then for every connected, semisimple group scheme G over K, H^{1}(Gal(K^{s}/K), G(K^{s})) is {∗}, and the converse holds. More generally, if K has cohomological dimension ≤ 1, then H^{1}(Gal(K^{s}/K), G(K^{s})) equals {∗} for every connected, reductive group scheme G over K. If K is perfect, it also holds for every connected, linear algebraic group scheme G over K.

If H^{1}(Gal(K^{s}/K), PGL_{n,K}(K^{s})) equals {∗} for every n, then Br′(K) equals {[K]} by Theorem 3.15. Using Proposition 3.8 and similar arguments, this implies cd(K) ≤ 1, cf. also Theorem 3.3. For a field of cohomological dimension ≤ 1, H^{1}(Gal(K^{s}/K), T) equals {∗} for every multiplicative group T, cf. [Ser79, Application, p. 162]. Thus Steinberg’s theorem reduces to the case that G is connected and semisimple. There is a further reduction to the case that G is simply connected and quasi-split, i.e., there exists a closed subgroup B of G such that B ⊗_{K} K^{s} is a maximal solvable subgroup (i.e., a Borel subgroup) of G ⊗_{K} K^{s}. Steinberg proves that for every field K, for every simply connected, semisimple group G with a Borel subgroup defined over K, every G-torsor arises as i_{∗}E where i : T → G is the inclusion of a multiplicative subgroup and where E is a T -torsor. When K has cohomological dimension ≤ 1, E is a trivial T -torsor, and thus also i_{∗}E is a trivial G-torsor.

Conjecture 5.5 (Serre’s “Conjecture II”). [Ser02, §3] If K has cohomological dimension 2 and if K is perfect, then for every connected, simply connected, semisimple group scheme G̃ over K, H^{1}(Gal(K^{s}/K), G̃(K^{s})) is {∗}. Equivalently, for every connected, semisimple group scheme G over K, the connecting map

Δ : H^{1}(Gal(K^{s}/K), G(K^{s})) → H_{fppf}^{2}(K; π_{1}G)

is injective.

Remark 5.6. In [Ser95], Serre explains that the hypothesis that K is perfect is too strong in these conjectures. They should also hold if the perfect hypothesis is replaced by the hypothesis that [K : K^{p}] ≤ p^{2} and H_{p}^{3}(K′) is 0 for all finite, separable extensions K′ of K. In particular, this new hypothesis holds when K is a function field of a surface over an algebraically closed field k, i.e., K is a finitely generated k-extension and tr.deg.(K/k) equals 2.

This conjecture remains open in general, although much is known, cf. [Ser02, §3], [Ser95], [CTGP04]. At least when K is perfect, a converse theorem is known, cf. Theorem 3.3(ii). We will return to this conjecture in Chapter 5.

CHAPTER 2 The Chevalley-Warning and Tsen-Lang theorems

1. The Chevalley-Warning Theorem

A motivating problem in both arithmetic and geometry is the following.

Problem 1.1. Given a field K and a K-variety X find sufficient, resp. necessary, conditions for existence of a K-point of X.

The problem depends dramatically on the type of K: number field, finite field, p-adic field, function field over a finite field, or function field over an algebraically closed field. In arithmetic the number field case is most exciting. However the geometric case, i.e., the case of a function field over an algebraically closed field, is typically easier and may suggest approaches and conjectures in the arithmetic case.

Two results, the Chevalley-Warning theorem and Tsen’s theorem, deduce a sufficient condition for existence of K-points by “counting”. More generally, counting leads to a relative result: the Tsen-Lang theorem that a strong property about existence of k-points for a field k propagates to a weaker property about K-points for certain field extensions K/k. The prototype result, both historically and logically, is a theorem of Chevalley and its generalization by Warning. The counting result at the heart of the proof is Lagrange’s theorem together with the observation that a nonzero single-variable polynomial of degree ≤ q − 1 cannot have q distinct zeroes.

Lemma 1.2. For a finite field K with q elements, the polynomial 1 − x^{q−1} vanishes on K^{∗} and x^{q} − x vanishes on all of K. For every integer n ≥ 0, for the K-algebra homomorphism

ev_{n} : K[X_{0}, …, X_{n}] → Hom_{Sets}(K^{n+1}, K),
ev_{n}(p(X_{0}, …, X_{n})) = ((a_{0}, …, a_{n}) ↦ p(a_{0}, …, a_{n})),

the kernel equals the ideal

I_{n} = ⟨X_{0}^{q} − X_{0}, …, X_{n}^{q} − X_{n}⟩.

Finally, the collection (X_{i}^{q} − X_{i})_{i=0,...,n} is a Gröbner basis with respect to every monomial order refining the grading of monomials by total order. In particular, for every p in I_{n} some term of p of highest degree is in the ideal ⟨X_{0}^{q}, …, X_{n}^{q}⟩.

Proof. Because K^{∗} is a group of order q − 1, Lagrange’s theorem implies a^{q−1} = 1 for every element a of K^{∗}, i.e., 1 − x^{q−1} vanishes on K^{∗}. Multiplying by x shows that x^{q} − x vanishes on K. Thus the ideal I_{n} is at least contained in the kernel of ev_{n}.

Modulo X_{n}^{q} − X_{n}, every element of K[X_{0}, …, X_{n}] is congruent to one of the form

p(X_{0}, …, X_{n}) = p_{q−1}X_{n}^{q−1} + ⋯ + p_{0}X_{n}^{0}, p_{0}, …, p_{q−1} ∈ K[X_{0}, …, X_{n−1}].

(Of course K^{n} is defined to be {0} and K[X_{0}, …, X_{n−1}] is defined to be K if n equals 0.) Since K has q elements and since a nonzero polynomial of degree ≤ q −1 can have at most q−1 distinct zeroes, for every (a_{0}, …, a_{n−1}) ∈ K^{n} the polynomial p(a_{0}, …, a_{n−1}, X_{n}) is zero on K if and only if

p_{0}(a_{0}, …, a_{n−1}) = ⋯ = p_{q−1}(a_{0}, …, a_{n−1}) = 0.

Thus ev_{n}(p) equals 0 if and only if each ev_{n−1}(p_{i}) equals 0. In that case, by the induction hypothesis, each p_{i} is in I_{n−1} (in case n = 0, each p_{i} equals 0). Then, since I_{n−1}K[X_{0}, …, X_{n}] is in I_{n}, p is in I_{n}. Therefore, by induction on n, the kernel of ev_{n} is precisely I_{n}.

Finally, Buchberger’s algorithm applied to the set (X_{0}^{q} − X_{0}, …, X_{n}^{q} − X_{n}) produces S-polynomials

S_{i,j} = X_{j}^{q}(X_{i}^{q} − X_{i}) − X_{i}^{q}(X_{j}^{q} − X_{j}) = X_{j}(X_{i}^{q} − X_{i}) − X_{i}(X_{j}^{q} − X_{j})

which have remainder 0. Therefore this set is a Gröbner basis by Buchberger’s criterion. □

Theorem 1.3. [Che35],[War35] Let K be a finite field. Let n and r be positive integers. Let F_{1}, …, F_{r} be nonconstant, homogeneous polynomials in K[X_{0}, …, X_{n}]. If

deg(F_{1}) + ⋯ + deg(F_{r}) ≤ n

then there exists (a_{0}, …, a_{n}) ∈ K^{n+1} − {0} such that for every i = 1, …, r, F_{i}(a_{0}, …, a_{n}) equals 0. Stated differently, the projective scheme 𝕍(F_{1}, …, F_{r}) ⊂ ℙ_{K}^{n} has a K-point.

Proof. Denote by q the number of elements in K. The polynomial

G(X_{0}, …, X_{n}) = 1 − ∏_{i=0}^{n}(1 − X_{i}^{q−1})

equals 0 on {0} and equals 1 on K^{n+1} − {0}. For the same reason, the polynomial

H(X_{0}, …, X_{n}) = 1 − ∏_{j=1}^{r}(1 − F_{j}(X_{0}, …, X_{n})^{q−1})

equals 0 on

{(a_{0}, …, a_{n}) ∈ K^{n+1}|F_{1}(a_{0}, …, a_{n}) = ⋯ = F_{r}(a_{0}, …, a_{n}) = 0}

and equals 1 on the complement of this set in K^{n+1}. Since each F_{i} is homogeneous, 0 is a common zero of F_{1}, …, F_{r}. Thus the difference G − H equals 1 on

{(a_{0}, …, a_{n}) ∈ K^{n+1} − {0}|F_{1}(a_{0}, …, a_{n}) = ⋯ = F_{r}(a_{0}, …, a_{n}) = 0}

and equals 0 on the complement of this set in K^{n+1}. Thus, to prove that F_{1}, …, F_{r} have a nontrivial common zero, it suffices to prove the polynomial G − H does not lie in the ideal I_{n}.

Since

deg(F_{1}) + ⋯ + deg(F_{r}) ≤ n,

H has strictly smaller degree than G. Thus the leading term of G − H equals the leading term of G. There is only one term of G of degree deg(G). Thus, for every monomial ordering refining the grading by total degree, the leading term of G equals

(−1)^{n+1}X_{0}^{q−1}X_{1}^{q−1} ⋯ X_{n}^{q−1}.

This is clearly divisible by none of X_{i}^{q} for i = 0, …, n, i.e., the leading term of G − H is not in the ideal ⟨X_{0}^{q}, …, X_{n}^{q}⟩. Because (X_{0}^{q} − X_{0}, …, X_{n}^{q} − X_{n}) is a Gröbner basis for I_{n} with respect to the monomial order, G − H is not in I_{n}. □

2. The Tsen-Lang Theorem

On the geometric side, an analogue of Chevalley’s theorem was proved by Tsen, cf. [Tse33]. This was later generalized independently by Tsen and Lang, cf. [Tse36], [Lan52]. Lang introduced a definition which simplifies the argument.

Definition 2.1. [Lan52] Let m be a nonnegative integer. A field K is called C_{m}, or said to have property C_{m}, if it satisfies the following. For every positive integer n and every sequence of positive integers (d_{1}, …, d_{r}) satisfying

d_{1}^{m} + ⋯ + d_{r}^{m} ≤ n,

every sequence (F_{1}, …, F_{r}) of homogeneous polynomials F_{i} ∈ K[X_{0}, …, X_{n}] with deg(F_{i}) = d_{i} has a common zero in K^{n+1} − {0}.

Remark 2.2. In fact the definition in [Lan52] is a little bit different than this. For fields having normic forms, Lang proves the definition above is equivalent to his definition. And the definition above works best with the following results.

With this definition, the statement of the Chevalley-Warning theorem is quite simple: every finite field has property C_{1}. The next result proves that property C_{m} is preserved by algebraic extension.

Lemma 2.3. For every nonnegative integer m, every algebraic extension of a field with property C_{m} has property C_{m}.

Proof. Let K be a field with property C_{m} and let L′/K be an algebraic extension. For every sequence of polynomials (F_{1}, …, F_{r}) as in the definition, the coefficients generate a finitely generated subextension L/K of L′/K. Thus clearly it suffices to prove the lemma for finitely generated, algebraic extensions L/K.

Denote by e the finite dimension dim_{K}(L). Because multiplication on L is K-bilinear, each homogeneous, degree d_{i}, polynomial map of L-vector spaces,

F_{i} : L^{⊕(n+1)} → L,

is also a homogeneous, degree d_{i}, polynomial map of K-vector spaces. Choosing a K-basis for L and decomposing F_{i} accordingly, F_{i} is equivalent to e distinct homogeneous, degree d_{i}, polynomial maps of K-vector spaces,

F_{i,j} : L^{⊕(n+1)} → K, j = 1, …, e.

The set of common zeroes of the collection of homogeneous polynomial maps (F_{i}|i = 1, …, r) equals the set of common zeroes of the collection of homogeneous polynomial functions (F_{i,j}|i = 1, …, r, j = 1, …, e). Thus it suffices to prove there is a nontrivial common zero of all the functions F_{i,j}.

By hypothesis,

∑_{i=1}^{r} deg(F_{i})^{m} is no greater than n.

Thus, also

∑_{i=1}^{r} ∑_{j=1}^{e} deg(F_{i,j})^{m} = e ∑_{i=1}^{r} deg(F_{i})^{m} is no greater than en.

Since K has property C_{m} and since

dim_{K}(L^{⊕(n+1)}), i.e., (n + 1)dim_{K}(L) = e(n + 1),

is larger than en, the collection of homogeneous polynomials F_{i,j} has a common zero in L^{⊕(n+1)} − {0}. □

The heart of the Tsen-Lang theorem is the following proposition.

Proposition 2.4. Let K/k be a function field of a curve, i.e., a finitely generated, separable field extension of transcendence degree 1. If k has property C_{m} then K has property C_{m+1}.

This is proved in a series of steps. Let n, r and d_{1}, …, d_{r} be positive integers such that

d_{1}^{m+1} + ⋯ + d_{r}^{m+1} ≤ n.

For every collection of homogeneous polynomials

F_{1}, …, F_{r} ∈ K[X_{0}, …, X_{n}], deg(F_{i}) = d_{i},

the goal is to prove that the collection of homogeneous, degree d_{i}, polynomial maps of K-vector spaces

F_{1}, …, F_{r} : K^{⊕(n+1)} → K

has a common zero. Of course, as in the proof of Lemma 2.3, this is also a collection of homogeneous polynomial maps of k-vector spaces. Unfortunately both of these k-vector spaces are infinite dimensional. However, using geometry, these polynomial maps can be realized as the colimits of polynomial maps of finite dimensional k-vector spaces. For these maps there is an analogue of the Chevalley-Warning argument replacing the counting argument by a parameter counting argument which ultimately follows from the Riemann-Roch theorem for curves. The first step is to give a projective model of K/k.

Lemma 2.5. For every separable, finitely generated field extension K/k of transcendence degree 1, there exists a smooth, projective, connected curve C over k and an isomorphism of k-extensions K ≅ k(C). Moreover the pair (C, K ≅ k(C)) is unique up to unique isomorphism.

Proof. This is essentially the Zariski-Riemann surface of the extension K/k. For a proof in the case that k is algebraically closed, see [Har77, Theorem I.6.9]. The proof in the general case is similar. □

The isomorphism K ≅ k(C) is useful because the infinite dimensional k-vector space k(C) has a plethora of naturally-defined finite dimensional subspaces. For every Cartier divisor D on C, denote by V_{D} the subspace

V_{D} := H^{0}(C, O_{C}(D)) = {f ∈ k(C)|div(f) + D ≥ 0}.

The collection of all Cartier divisors D on C is a partially ordered set where

D′ ≥ D if and only if D′ − D is effective.

The system of subspaces V_{D} of k(C) is compatible for this partial order, i.e., if D′ ≥ D then V_{D′} ⊃ V_{D}. And K is the union of all the subspaces V_{D}, i.e., it is the colimit of this compatible system of finite dimensional k-vector spaces. Thus for all k-multilinear algebra operations which commute with colimits, the operation on k(C) can be understood in terms of its restrictions to the finite dimensional subspaces k(C). The next lemma makes this more concrete for the polynomial map F.

Lemma 2.6. Let C be a smooth, projective, connected curve over a field k and let

F_{i} ∈ k(C)[X_{0}, …, X_{n}]_{d_{i}}, i = 1, …, r

be a collection of polynomials in the spaces k(C)[X_{0}, …, X_{n}]_{d_{i}} of homogeneous, degree d_{i} polynomials. There exists an effective, Cartier divisor P on C and for every i = 1, …, r there exists a global section F_{C,i} of the coherent sheaf O_{C}(P)[X_{0}, …, X_{n}]_{d_{i}} such that for every i = 1, …, r the germ of F_{C,i} at the generic point of C equals F_{i}.

Remark 2.7. In particular, for every Cartier divisor D on C and for every i = 1, …, r there is a homogeneous, degree d, polynomial map of k-vector spaces

F_{C,D,i} : V_{D}^{⊕(n+1)} → W_{d_{i},P,D}, W_{d_{i},P,D} := V_{d_{i}D+P} ,

such that for every i = 1, …, r the restriction of F_{i} to V_{D}^{⊕(n+1)} equals F_{C,D,i} considered as a map with target K (rather than the subspace V_{dD+P}).

Proof. The coefficients of each F_{i} are rational functions on C. Each such function has a polar divisor. Since there are only finitely many coefficients of the finitely many polynomials F_{1}, …, F_{r}, there exists a single effective, Cartier divisor P on C such that every coefficient is a global section of O_{C}(P). □

Because of Lemma 2.6, the original polynomial maps F_{1}, …, F_{r} can be understood in terms of their restrictions to the subspaces V_{D}. The dimensions of these subspaces are determined by the Riemann-Roch theorem.

Theorem 2.8 (Riemann-Roch for smooth, projective curves). Let k be a field. Let C be a smooth, projective, connected curve over k. Denote by ω_{C/k} the sheaf of relative differentials of C over k and denote by g(C) = genus(C) the unique integer such that deg(ω_{C/k}) = 2g(C) − 2. For every invertible sheaf L on C,

h^{0}(C, L) − h^{0}(C, ω_{C} ⊗_{O_{C}} L^{∨}) = deg(L) + 1 − g(C).

Remark 2.9. In particular, if deg(L) > deg(ω_{C}) = 2g(C)−2 so that ω_{C}⊗_{O_{C}} L^{∨} has negative degree, then h^{0}(C, ω_{C} ⊗_{O_{C}} L^{∨}) equals zero. And then

h^{0}(C, L) = deg(L) + 1 − g(C).

For a Cartier divisor D satisfying

deg(D) > 2g(C) − 2 and for each i = 1, …, r, d_{i}deg(D) + deg(P) > 2g(C) − 2,

the Riemann-Roch theorem gives that V_{D}^{⊕(n+1)} and W_{d_{i},P,D} are finite dimensional k-vector spaces of respective dimensions,

dim_{k}(V_{D}^{⊕(n+1)}) = (n + 1)h^{0}(C, O_{C}(D)) = (n + 1)(deg(D) + 1 − g)

and

dim_{k}(W_{d_{i},P,D}) = dim(V_{d_{i}D+P}) = d_{i}deg(D) + deg(P) + 1 − g.

In this case, choosing a basis for W_{d_{i},P,D} and decomposing

F_{C,D,i} : V_{D}^{⊕(n+1)} → W_{d_{i},P,D}

into its associated components, there exist dim_{k}(W_{d_{i},P,D}) homogeneous, degree d, polynomial functions

(F_{C,D,i})_{j} : V_{D}^{⊕(n+1)} → k, j = 1, …, dim_{k}(W_{d_{i},P,D})

such that a zero of F_{C,D,i} is precisely the same as a common zero of all the functions (F_{C,D,i})_{j}.

Proof of Proposition 2.4. By hypothesis, each d_{i} and n + 1 − ∑_{i=1}^{r} d_{i}^{m+1} are nonzero so that the fractions

\frac{2g(C) − 2 − deg(P)}{d_{i}} for each i = 1, …, r, \frac{(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)}{n + 1 − ∑_{i=1}^{r} d_{i}^{m+1}}

are all defined. Let D be an effective, Cartier divisor on C such that

deg(D) > 2g(C) − 2, deg(D) > \frac{2g(C) − 2 − deg(P)}{d_{i}}, i = 1, …, r, and

deg(D) > \frac{(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)}{n + 1 − ∑_{i=1}^{r} d_{i}^{m+1}}.

Because deg(D) > 2g(C) − 2, the Riemann-Roch theorem states that

dim_{k}(V_{D}^{⊕(n+1)}) = (n + 1)dim_{k}(V_{D}) = (n + 1)(deg(D) + 1 − g).

For every i = 1, …, r, because d_{i} is positive and because deg(D) > (2g(C) − 2 − deg(P))/d_{i}, also

deg(d_{i}D + P) = d_{i}deg(D) + deg(P) is greater than 2g(C) − 2.

Thus the Riemann-Roch theorem states that

dim_{k}(W_{d_{i},P,D}) = dim_{k}(V_{d_{i}D+P}) = d_{i}deg(D) + deg(P) + 1 − g(C).

Thus for the collection of polynomial functions (F_{C,D,i})_{j},

dim_{k}(V_{D}^{⊕(n+1)}) − ∑_{i=1}^{r} ∑_{j} deg((F_{C,D,i})_{j})^{m}

equals

(n + 1)(deg(D) + 1 − g) − ∑_{i=1}^{r}(d_{i}deg(D) + deg(P) + 1 − g(C))d_{i}^{m} =

(n + 1 − ∑_{i=1}^{r} d_{i}^{m+1})deg(D) − [(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)].

Because

deg(D) > \frac{(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)}{n + 1 − ∑_{i=1}^{r} d_{i}^{m+1}}

and because n + 1 − ∑_{i=1}^{r} d_{i}^{m+1} is positive, also

(n + 1 − ∑_{i=1}^{r} d_{i}^{m+1})deg(D) > [(n + 1 − ∑_{i=1}^{r} d_{i}^{m})(g − 1) + ∑_{i=1}^{r} d_{i}^{m}deg(P)].

Therefore

dim_{k}(V_{D}^{⊕(n+1)}) is greater than ∑_{i=1}^{r} ∑_{j} deg((F_{i,C,D})_{j})^{m}.

Because of the inequality above, and because k has property C_{m}, there is a nontrivial common zero of the collection of homogeneous polynomial functions (F_{C,D,i})_{j}, i = 1, …, r, j = 1, …, dim_{k}(W_{d_{i},P_{i},D}). Therefore there is a nontrivial common zero of the collection of homogeneous polynomial maps F_{C,D,i}, i = 1, …, r. By Lemma 2.6, the image of this nonzero element in K^{⊕(n+1)} is a nonzero element which is a common zero of the polynomials F_{1}, …, F_{r}. □

Proposition 2.4 is the main step in the proof of the Tsen-Lang theorem.

Theorem 2.10 (The Tsen-Lang Theorem). [Lan52] Let K/k be a field extension with finite transcendence degree, tr.deg.(K/k) = t. If k has property C_{m} then K has property C_{m+t}.

Proof. The proof of the theorem is by induction on t. When t = 0, i.e., when K/k is algebraic, the result follows from Lemma 2.3. Thus assume t > 0 and the result is known for t − 1. Let (b_{1}, …, b_{t}) be a transcendence basis for K/k. Let E_{t}, resp. E_{t−1}, denote the subfield of K generated by k and b_{1}, …, b_{t}, resp. generated by k and b_{1}, …, b_{t−1}. Since E_{t−1}/k has transcendence degree t − 1, by the induction hypothesis E_{t−1} has property C_{m+t−1}. Now E_{t}/E_{t−1} is a purely transcendental extension of transcendence degree 1. In particular, it is finitely generated and separable. Since E_{t−1} has property C_{m+t−1}, by Proposition 2.4 E_{t} has property C_{m+t}. Finally by Lemma 2.3 again, since K/E_{t} is algebraic and E_{t} has property C_{m+t}, also K has property C_{m+t}. □

The homogeneous version of the Nullstellensatz implies a field k has property C_{0} if and only if k is algebraically closed. Thus one corollary of Theorem 2.10 is the following.

Corollary 2.11. Let k be an algebraically closed field and let K/k be a field extension of finite transcendence degree t. The field K has property C_{t}.

In particular, the case t = 1 is historically the first result in this direction.

Corollary 2.12 (Tsen’s theorem). [Tse36] The function field of a curve over an algebraically closed field has property C_{1}.

3. Applications to Brauer groups

Chevalley and Tsen recognized that property C_{1}, which they called quasialgebraic closure, has an important consequence for division algebras. Lang recognized that property C_{2} also has an important consequence for division algebras, cf. [Lan52, Theorem 13].

Let A be a central simple algebra over K with dim_{K}(A) = n^{2}, and let E be the corresponding PGL_{n,K}-torsor. Because the determinant polynomial det on Mat_{n×n,K} is invariant for the inner action of PGL_{n,K}, det determines a well-defined polynomial map on A.

Definition 3.1. For a central simple algebra A over K with dim_{K}(A) = n^{2}, the reduced norm is the unique, degree n polynomial map

Nrm_{A/k} : A → K

such that the induced degree n polynomial map

Nrm_{A/K} ⊗ Id : A ⊗_{K} K^{s} → K^{s}

agrees with the determinant map

det : Mat_{n×n,K^{s}} → K^{s}

via one, and hence every, K^{s}-algebra isomorphism of A and Mat_{n×n,K^{s}}.

Since det is multiplicative, so is Nrm_{A/K}, i.e.,

∀a, b ∈ A, Nrm_{A/K}(ab) = Nrm_{A/K}(a)Nrm_{A/K}(b).

And the restriction to the center K is the polynomial map λ ↦ λ^{n}. These properties characterize the reduced norm. By the same type of Galois invariance argument as above, and using Cramer’s rule, an element a of A has a (left and right) inverse if and only if Nrm_{A/K}(a) is nonzero. In particular, if D is a division algebra the only zero of Nrm_{A/K} is a = 0.

Proposition 3.2. Let K be a field

(i) If K has property C_{1}, then the only division algebra with center K is K itself. Thus Br(K) equals {[K]}.

(ii) If K has property C_{2} then for every division algebra D with center K the reduced norm map

Nrm_{D/K} : D → K

is surjective.

Proof. (i). Let D be a division algebra with center K. Denote by n the index of D. Because Mat_{n×n}(\overline{K}) has dimension n^{2} as a \overline{K}-vector space, also D has dimension n^{2} as a K-vector space. If K has property C_{1}, then since the homogeneous polynomial map Nrm_{D/K} has only the trivial zero,

n = deg(Nrm_{D/K}) ≥ dim_{K}(D) = n^{2},

i.e., n = 1. Thus for a field K with property C_{1}, the only finite dimensional, division algebra with center K has dimension 1, i.e., D equals K.

(ii). Next suppose that K has property C_{2}. Clearly Nrm_{D/k}(0) equals 0. Thus to prove that

Nrm_{D/K} : D → K

is surjective, it suffices to prove that for every nonzero c ∈ K there exists b in D with Nrm_{D/K}(b) = c. Consider the homogeneous, degree n, polynomial map

F_{c} : D ⊕ K → K, (a, λ) ↦ Nrm_{D/K}(a) − cλ^{n}.

Since

dim_{K}(D ⊕ K) = n^{2} + 1 > deg(F_{c})^{2},

by property C_{2} the map F_{c} has a zero (a, λ) ≠ (0, 0), i.e., Nrm_{D/k}(a) = cλ^{n}. In particular, λ must be nonzero since otherwise a is a nonzero element of D with Nrm_{D/K}(a) = 0. But then b = (1/λ)a is an element of D with Nrm_{D/k}(b) = c. □

It was later recognized, particularly through the work of Merkurjev and Suslin, that these properties of division algebras are equivalent to properties of Galois cohomology.

Theorem 3.3. [Ser02, Proposition 5, §I.3.1], [Sus84, Corollary 24.9] Let K be a field.

(i) The cohomological dimension of K is ≤ 1 if and only if for every finite extension L/K, the only division algebra with center L is L itself.

(ii) If K is perfect, the cohomological dimension of K is ≤ 2 if and only if for every finite extension L/K, for every division algebra D with center L, the reduced norm map Nrm_{D/L} is surjective.

CHAPTER 3 Rationally connected fibrations over curves

1. Rationally connected varieties

The theorems of Chevalley-Warning and Tsen-Lang are positive answers to Problem 1.1 for certain classes of fields. It is natural to ask whether these theorems can be generalized for such fields.

Problem 1.1. Let r be a nonnegative integer. Give sufficient geometric conditions on a variety such that for every C_{r} field K (or perhaps every C_{r} field satisfying some additional hypotheses) and for every K-variety satisfying the conditions, X has a K-point.

As with Problem 1.1, this problem is quite vague. Nonetheless there are important partial answers. One such answer is the following.

Theorem 1.2. [Man86] [CT87] Let K be a C_{1} field and let X be a projective K-variety. If X ⊗_{K} \overline{K} is birational to ℙ2 ; \overline{K} then X has a K-point.

This begs the question: What (if anything) is the common feature of rational surfaces and of the varieties occurring in the Chevalley-Warning and Tsen-Lang theorems, i.e., complete intersections in ℙ^{n} of hypersurfaces of degrees d_{1}, …, d_{r} with d_{1} + ⋯ + d_{r} ≤ n? One answer is rational connectedness. This is a property that was studied by Kollár-Miyaoka-Mori and Campana, cf. [Kol96].

Definition 1.3. Let k be an algebraically closed field. An integral (thus nonempty), separated, finite type, k-scheme X is rationally connected, resp. separably rationally connected, if there exists an integral, finite type k-scheme M and a morphism of k-schemes

u : M ×_{k} ℙ_{k}^{1} → X, (m, t) ↦ u(m, t)

such that the induced morphism of k-schemes

u^{(2)} : M ×_{k} ℙ_{k}^{1} ×_{k} ℙ_{k}^{1} → X ×_{k} X, (m, t_{1}, t_{1}) ↦ (u(m, t_{1}), u(m, t_{2}))

is surjective, resp. surjective and generically smooth.

In a similar way, X is rationally chain connected, resp. separably rationally chain connected, if for some integer m ≥ 1, the analogous property holds after replacing ℙ_{k}^{1} by the proper, connected, nodal, reducible curve C_{m} which is a chain of m smooth rational curves.

In other references, the condition on u^{(2)} is that it be dominant rather than surjective. In the following arguments, it is preferrable to demand that u^{(2)} is surjective. When X is proper and char(k) = 0, the definitions turn out to be equivalent.

Figure 1. Every pair of points in a rationally connected variety lie in an image of the projective line

Figure 1 shows a rationally connected variety, where every pair of points is contained in an image u(ℙ^{1}) of the projective line.

The definition of rational connectedness, resp. rational chain connectedness, mentions a particular parameter space M. However, using the general theory of Hilbert schemes, it suffices to check that every pair (x_{1}, x_{2}) of K-points of X ⊗_{k} K is contained in some rational K-curve, resp. a chain of rational K-curves, (not necessarily from a fixed parameter space) for one sufficiently large, algebraically closed, field extension K/k, i.e., for an algebraically closed extension K/k such that for every countable collection of proper closed subvarieties Y_{i} ⊊ X, there exists a K-point of X contained in none of the sets Y_{i}. For instance, K/k is sufficiently large if K is uncountable or if K/k contains the fraction field k(X)/k as a subextension.

A very closely related property is the existence of a very free rational curve. For a d-dimensional variety X, a very free rational curve is a morphism

f : ℙ_{k}^{1} → X_{smooth}

into the smooth locus of X such that f^{∗}T_{X} is ample, i.e.,

f^{∗}T_{X} ≅ O_{ℙ_{k}^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ_{k}^{1}}(a_{d}), a_{1}, …, a_{d} > 0.

Definition 1.4. Let k be a field and let X be a quasi-projective k-scheme. Denote by X_{smooth} the smooth locus of X. The very free locus X_{v.f.} of X is the union of the images in X_{smooth} of all very free rational curves to X_{smooth} ⊗_{k} K as K/k varies over all algebraically closed extensions. More generally, for a flat, quasi-projective morphism of schemes,

π : X → B,

denoting by X_{π,smooth} the smooth locus of the morphism π, the very free locus X_{π,v.f.} is the union in X_{π,smooth} of the images of every very free rational curve in every geometric fiber of X_{π,smooth} over B.

The next theorem explains the relation of these different properties.

Theorem 1.5. [Kol96, §IV.3], [HT06], Unless stated otherwise, all varieties below are d-dimensional, reduced, irreducible, quasi-projective schemes over an algebraically closed field k.

(0) In characteristic 0, every rationally connected variety is separably rationally connected.

(i) For every flat, proper morphism π : X → B (not necessarily of quasiprojective varieties over a field), the subset of B parameterizing points whose geometric fiber is rationally chain connected is stable under specialization. (If one bounds the degree of the chains with respect to a relatively ample invertible O_{X}-module, then it is a closed subset.)

(ii) The very free locus of a quasi-projective variety is open. More generally, for every flat, quasi-projective morphism, π : X → B, the subset X_{π,v.f.} of X_{π,smooth} is an open subset.

(iii) The very free locus X_{v.f.} of a quasi-projective variety is (separably) rationally connected in the following strong sense. For every positive integer N, for every positive integer m, and for every positive integer a, for every collection of distinct closed points t_{1}, …, t_{N} ∈ ℙ_{k}^{1}, for every collection of closed points x_{1}, …, x_{N} ∈ X_{v.f.}, and for every specification of an m-jet of a smooth curve in X at each point x_{i}, there exists a morphism

f : ℙ_{k}^{1} → X_{v.f.}

such that for every i = 1, …, N, f is unramified at t_{i}, f(t_{i}) equals x_{i} and the m-jet of t_{i} in ℙ_{k}^{1} maps isomorphically to the specified m-jet at x_{i}, and

f^{∗}T_{X} ≅ O_{ℙ_{k}^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ_{k}^{1}}(a_{d}), a_{1}, …, a_{d} ≥ a.

(iv) Every rational curve in X_{smooth} intersecting X_{v.f.} is contained in X_{v.f.}. Thus for every smooth, rationally chain connected variety, if X contains a very free rational curve then X_{v.f.} equals all of X.

(v) A proper variety X is rationally chain connected if it is generically rationally chain connected, i.e., if there exists a morphism u as in the definition such that u^{(2)} is dominant (but not necessarily surjective).

(vi) For the morphism u : M ×_{k} ℙ_{k}^{1} → X, let l be a closed point of M such that u_{l} : ℙ^{1} → X has image in X_{smooth} and such that u^{(2)} is smooth at (l, t_{1}, t_{2}) for some t_{1}, t_{2} ∈ ℙ_{k}^{1}. Then the morphism u_{l} is very free. Thus an irreducible, quasi-projective variety X contains a very free curve if and only if there is a separably rationally connected open subset of X_{smooth}. Also, a smooth, quasi-projective variety X in characteristic 0 which is generically rationally connected contains a very free morphism.

(vii) For a surjective morphism f : X → Y of varieties over an algebraically closed field, if X is rationally connected, resp. rationally chain connected, then also Y is rationally connected, resp. rationally chain connected.

(viii) For a birational morphism f : X → Y of proper varieties over an algebraically closed field, if Y is rationally connected then X is rationally chain connected. If the characteristic is zero, then X is rationally connected.

Remark 1.6. Item (ii) is proved in Proposition 3.6. The generic case of Item (iii), which is all we will need, is proved in Proposition 3.7. The complete result was proved by Hassett and Tschinkel, [HT06]. Item (iv) follows from Corollary 3.8. The remaining items are not proved, nor are they used in the proof of the main theorem. For the most part they are proved by similar arguments; complete proofs are in [Kol96, §IV.3].

Rational connectedness is analogous to path connectedness in topology, and satisfies the analogues of many properties of path connectedness. One property of path connectedness is this: for a fibration of CW complexes, if the base space and the fibers are path connected, then also the total space is path connected. This led to two conjectures by Kollár, Miyaoka and Mori.

Conjecture 1.7. [Kol96, Conjecture IV.5.6] Let π : X → B be a surjective morphism of smooth, projective schemes over an algebraically closed field of characteristic 0. If both B and a general fiber of π are rationally connected, then X is also rationally connected.

Conjecture 1.7 is implied by the following conjecture about rationally connected fibrations over curves.

Conjecture 1.8. [Kol96, Conjecture IV.6.1.1] Let π : X → B be a surjective morphism of projective schemes over an algebraically closed field of characteristic 0. If B is a smooth curve and if a general fiber of f is rationally connected, then there exists a morphism s : B → X such that π ∘ s equals Id_{B}, i.e., s is a section of π.

Our next goal is to prove the following result.

Theorem 1.9. [GHS03] Conjecture 1.8 of Kollár-Miyaoka-Mori is true. Precisely, let k be an algebraically closed field of characteristic 0 and let π : X → B be a surjective morphism from a normal, projective k-scheme X to a smooth, projective, connected k-curve B. If the geometric generic fiber X_{\overline{η}_{B}} is a normal, integral scheme whose smooth locus contains a very free curve, then there exists a morphism s : B → X such that π ∘ s equals Id_{B}.

This was generalized by A. J. de Jong to the case that k is algebraically closed of arbitrary characteristic, [dJS03]. The key difference has to do with extensions of valuation rings in characteristic 0 and in positive characteristic. Given a flat morphism of smooth schemes in characteristic 0, π : U → B, and given codimension 1 points η_{D} of U and η_{Δ} of B with π(η_{D}) = η_{Δ}, the induced local homomorphism of stalks π_{U}^{∗} : Ô_{B,η_{Δ}} → Ô_{U,η_{D}}, is equivalent to

k(Δ) \begin{matrix}[ \\ [ & t & ] \\ ] \end{matrix} → k(D) \begin{matrix}[ \\ [ & r & ] \\ & ] \end{matrix} , t ↦ ur^{m}

for a unit u and a positive integer m, cf. the proof of Lemma 4.3 below. In particular, it is rigid in the sense that t ↦ ur^{m} + vr^{m+1} + … is equivalent to t ↦ ur^{m}. However, extensions of positive characteristic valuation rings are not rigid, e.g., t ↦ r^{p} + v_{1}r^{p+1} is equivalent to t ↦ r^{p} + v_{2}r^{p+1} only if v_{1} = v_{2}. But there is a weak rigidity of local homomorphisms, Krasner’s lemma in the theory of non-Archimedean valuations. This is a key step in the generalization to positive characteristic.

Of course when k has characteristic 0, then since X is normal the fiber X_{\overline{η}_{B}} is automatically normal. If X is also smooth (which can be achieved thanks to resolution of singularities in characteristic 0), then also X_{\overline{η}_{B}} is smooth. Then the hypothesis on X_{\overline{η}_{B}} is equivalent to rational connectedness.

2. Outline of the proof

The proof that follows is based on a proof by T. Graber, J. Harris and myself (not quite the version we chose to publish) together with several major simplifications due to A. J. de Jong. The basic idea is to choose a smooth curve C ⊂ X such that π|_{C} : C → B is finite, and then try to deform C as a curve in X until it specializes to a reducible curve in X, one component of which is the image of a section s of π. Here are some definitions that make this precise.

Definition 2.1. Let π_{C} : C → B be a finite morphism of smooth, projective k-curves. A linked curve with handle C is a reduced, connected, projective curve C_{link} with irreducible components

C_{link} = C ∪ L_{1} ∪ ⋯ ∪ L_{m}

together with a morphism

π_{C,link} : C_{link} → B

such that

(i) π_{C,link} restricts to π_{C} on the component C, (ii) the restriction of π_{C,link} to each link component L_{i} is a constant morphism

with image b_{i}, where b_{1}, …, b_{m} are distinct closed points of B, (iii) and each link L_{i} is a smooth, rational curve intersecting C in a finite

number of nodes of C_{link}.

If every link L_{i} intersects C in a single node of C_{link}, then (C_{link}, π_{C,link}) is called a comb and the links L_{i} are called teeth. For combs we will use the notation C_{comb} rather than C_{link}.

A one-parameter deformation of a linked curve (C_{link}, π_{C,link}) is a datum of a smooth, connected, pointed curve (Π, 0) and a projective morphism

(ρ, π_{C}) : C → Π ×_{k} B

such that ρ is flat and such that C_{0} := ρ^{−1}(0) together with the restriction of π_{C} equals the linked curve (C_{link}, π_{C,link}).

A one-parameter deformation specializes to a section curve if there exists a closed point ∞ ∈ Π and an irreducible component B_{i} of C_{∞} := ρ^{−1}(∞) such that

(i) C_{∞} is reduced at the generic point of B_{i} (ii) and the restriction of π_{C} to B_{i} is an isomorphism

π_{C}|_{B_{i}} : B_{i} ⟶^{≅} B.

Figure 2. A linked curve with handle C and some links intersecting C in more than 1 point

Given a linked curve, a one-parameter deformation of the linked curve and a B-morphism j : C_{link} → X, an extension of j is an open neighborhood of 0, 0 ∈ N ⊂ B and a B-morphism

j_{N} : C_{N} → X, C_{N} := ρ^{−1}(N)

restricting to j on C_{0} = C_{link}.

Figure 2 shows a linked curve with some links intersecting the handle in more than 1 point. And Figure 3 shows a comb, where every tooth intersects the handle precisely once.

For the purposes of producing a section, the particular parameter space (Π, 0) of the one-parameter deformation is irrelevant. Thus, it is allowed to replace the one-parameter deformation by the new one-parameter deformation obtained from a finite base change (Π′, 0′) → (Π, 0). The following lemma is straightforward.

Lemma 2.2. Let (Π, 0, ∞) together with (ρ, π_{C}) : C → Π×_{k}B be a one-parameter deformation of (C_{link}, π_{C,link}) specializing to a section curve B_{i}. For every morphism of 2-pointed, smooth, connected curves

(Π′, 0′, ∞′) → (Π, 0, ∞),

the base change morphism

Π′ ×_{Π} C → Π′ ×_{k} B

is also a one-parameter deformation of (C_{link}, π_{C,link}) specializing to the section curve B_{i}.

Figure 3. A comb is a linked curve where every link, or tooth, intersects the handle precisely once

The usefulness of these definitions is the following simple consequence of the valuative criterion of properness.

Lemma 2.3. Let (C_{link}, π_{link}) be a linked curve together with a B-morphism j : C_{link} → X. If there exists a one-parameter deformation of the linked curve specializing to a section curve and if there exists an extension of j, then there exists a section s : B → X of π.

Proof. Let R denote the stalk O_{C,η_{B_{i}}} of O_{C} at the generic point η_{B_{i}} of B_{i}. By the hypotheses on C and B_{i}, R is a discrete valuation ring with residue field κ = k(B_{i}) and fraction field K = k(C). The restriction of j_{N} to the generic point of C is a B-morphism

j_{K} : Spec K → X.

Because π : X → B is proper, by the valuative criterion of properness the Bmorphism j_{K} extends to a B-morphism

j_{R} : Spec R → X,

which in turn gives a B-morphism from the residue field Spec κ to X, i.e., a rational B-map

j_{B_{i}} : B_{i} ⊃ U → X, U ⊂ B_{i} a dense, Zariski open.

Finally, because B_{i} is a smooth curve, the valuative criterion applies once more and this rational transformation extends to a B-morphism

j_{B_{i}} : B_{i} → X.

Because π_{C}|_{B_{i}} : B_{i} → B is an isomorphism, there exists a unique B-morphism

s : B → X

such that j_{B_{i}} = s ∘ π_{C}|_{B_{i}}. The morphism s is a section of π. □

Thus the proof of the theorem breaks into three parts:

(i) find a “good” linked curve j : C_{link} → X, (ii) find a one-parameter deformation of the linked curve specializing to a

section curve, (iii) and find an extension of j to the one-parameter deformation. The first step in finding j : C_{link} → X is to form a curve C_{init} which is an

intersection of X with dim(X) − 1 general hyperplanes in projective space. By Bertini’s theorem, if the hyperplanes are sufficiently general, then C_{init} will satisfy any reasonable transversality property. Moreover, there is a technique due to Kollár-Miyaoka-Mori – the smoothing combs technique – for improving C_{init} to another curve C ⊂ X still satisfying the transversality property and also satisfying a positivity property with respect to the vertical tangent bundle of π : X → B.

Unfortunately, even after such an improvement, there may be no one-parameter deformation of π|_{C} : C → B specializing to a section curve. However, after attaching sufficiently many link components over general closed points of B, there does exist a one-parameter deformation of C_{link} specializing to a section curve. This is one aspect of the well-known theorem that for a fixed base curve B and for a fixed degree d, if the number β of branch points is sufficiently large the Hurwitz scheme of degree d covers of B with β branch points is irreducible. (This was proved by Hurwitz when g(B) = 0, [Hur91], proved by Richard Hamilton for arbitrary genus in his thesis, and periodically reproved ever since, cf. [GHS02].) Because the general fibers of π : X → B are rationally connected, the inclusion C ⊂ X extends to a B-morphism j : C_{link} → X.

Finally the positivity property mentioned above implies j extends to the oneparameter deformation, at least after base change by a morphism Π′ → Π.

3. Hilbert schemes and smoothing combs

The smoothing combs technique of Kollár-Miyaoka-Mori depends on a result from the deformation theory of Hilbert schemes. Here is the setup for this result. Let Y → S be a flat, quasi-projective morphism and let

(ρ_{Hilb} : Hilb(Y/S) → S, Univ(Y/S) ⊂ Hilb(Y/S) ×_{S} T)

be universal among pairs (ρ : T → S, Z ⊂ T ×_{S} Y) of an S-scheme T and a closed subscheme Z ⊂ T ×_{S} Y such that Z → T is proper, flat and finitely presented. In other words, Hilb(Y/S) is the relative Hilbert scheme of Y over S.

In particular, for every field K the K-valued points of Hilb(Y/S) are naturally in bijection with pairs (s, Z) of a K-valued point s of S and a closed subscheme Z of Y_{s} := {s} ×_{S} Y. The closed immersion Z → Y_{s} is a regular embedding if at every point of Z the stalk of the ideal sheaf I_{Z/Y_{s}} is generated by a regular sequence of elements in the stalk of O_{Y_{s}}. In this case the conormal sheaf I_{Z/Y_{s}}/I_{Z/Y_{s}}^{2} is a locally free O_{Z}-module, and hence also the normal sheaf

N_{Z/Y_{s}} := Hom_{O_{Z}} (I_{Z/Y_{s}}/I_{Z/Y_{s}}^{2}, O_{S})

is a locally free O_{Z}-module. The regular embeddings which arise in the proof of Theorem 1.9 are precisely closed immersions of at-worst-nodal curves in a smooth variety.

Proposition 3.1. [Kol96, Theorem I.2.10, Lemma I.2.12.1, Proposition I.2.14.2] If Z ⊂ Y_{s} is a regular embedding and if h^{1}(Z, N_{Z/Y_{s}}) equals 0, then Hilb(Y/S) is smooth over S at (s, Z).

There is a variation of this proposition which is also useful. There is a flag Hilbert scheme of Y over S, i.e., a universal pair

(ρ_{fHilb} : fHilb(Y/S) → S, Univ_{1}(Y/S) ⊂ Univ_{2}(Y/S) ⊂ Hilb(Y/S) ×_{S} T)

among all pairs (ρ : T → S, Z_{1} ⊂ Z_{2} ⊂ T ×_{S} Y) of an S-scheme T and a nested pair of closed subschemes Z_{1} ⊂ Z_{2} ⊂ T ×_{S} Y such that for i = 1, 2, the projection Z_{i} → T is proper, flat and finitely presented. There are obvious forgetful morphisms

F_{i} : fHilb(Y/S) → Hilb(Y/S), F_{i}(s, Z_{1}, Z_{2}) = (s, Z_{i}).

Proposition 3.2. Let K be a field and let (s, Z_{1}, Z_{2}) be a K-point of fHilb(Y/S). If each closed immersion Z_{1} ⊂ Z_{2} and Z_{2} ⊂ Y_{s} is a regular embedding and if

h^{1}(Z_{2}, N_{Z_{2}/Y_{s}}) = 0, h^{1}(Z_{1}, N_{Z_{1}/Z_{2}}) = 0, and if h^{i}(Z_{2}, I_{Z_{1}/Z_{2}}·N_{Z_{2}/Y_{2}}) = 0 for i = 1, 2,

then fHilb(Y/S) is smooth over S at (s, Z_{1}, Z_{2}), for each i = 1, 2, Hilb(Y/S) is smooth over S at (s, Z_{i}), and each forgetful morphism F_{i} : fHilb(Y/S) → Hilb(Y/S) is smooth at (s, Z_{1}, Z_{2}).

Proof. Since h^{1}(Z_{2}, N_{Z_{2}/Y_{s}}) equals 0, Hilb(Y/S) is smooth at (s, Z_{2}) by Proposition 3.1. It is easy to see that the forgetful morphism F_{2} is equivalent to the relative Hilbert scheme Hilb(Univ(Y/S)/Hilb(Y/S)) over Hilb(Y/S). Thus, applying Proposition 3.1 to this Hilbert scheme, the vanishing of h^{1}(Z_{1}, N_{Z_{1}/Z_{2}}) implies F_{2} is smooth at (s, Z_{1}, Z_{2}). Since a composition of smooth morphisms is smooth, also fHilb(Y/S) is smooth over S at (s, Z_{1}, Z_{2}). The long exact sequence of cohomology associated to the short exact sequence

0 ⟶ I_{Z_{1}/Z_{2}} · N_{Z_{2}/Y_{s}} ⟶ N_{Z_{2}/Y_{s}} ⟶ N_{Z_{2}/Y_{s}}|_{Z_{1}} ⟶ 0

implies that h^{1}(Z_{1}, N_{Z_{2}/Y_{s}}|_{Z_{1}}) equals h^{1}(Z_{2}, N_{Z_{2}/Y_{s}}), which is 0. Thus, the long exact sequence of cohomology associated to

0 ⟶ N_{Z_{1}/Z_{2}} ⟶ N_{Z_{1}/Y_{s}} ⟶ N_{Z_{2}/Y_{s}}|_{Z_{1}} ⟶ 0

implies that h^{1}(Z_{2}, N_{Z_{1}/Y_{2}}) equals 0. So again by Proposition 3.1, Hilb(Y/S) is smooth over S at (s, Z_{1}). Finally, F_{1} is a morphism of smooth S-schemes at (s, Z_{1}, Z_{2}). Thus, to prove F_{1} is smooth, it suffices to prove it is surjective on Zariski tangent vector spaces. This follows from the vanishing of h^{1}(Z_{2}, I_{Z_{1}/Z_{2}} · N_{Z_{2}/Y_{s}}). □

Another ingredient in the smoothing combs technique is a a simple result about elementary transforms of locally free sheaves on a curve: the higher cohomology of the sheaf becomes zero after applying elementary transforms at sufficiently many points.

Lemma 3.3. Let C be a projective, at-worst-nodal, connected curve over a field k and let E be a locally free O_{C}-module.

(i) There exists a short exact sequence of coherent sheaves,

0 → F^{∨} → E^{∨} → T → 0

such that T is a torsion sheaf with support in C_{smooth} and such that h^{1}(C, F) equals 0.

(ii) Inside the parameter space of torsion quotients q : E^{∨} ↠ T with support in C_{smooth}, denoting

F^{∨} := Ker(E^{∨} → T ) and F := Hom_{O_{C}} (F^{∨}, O_{C}),

the subset parameterizing quotients for which h^{1}(C, F) = 0 is an open subset.

(iii) If h^{1}(C, F) equals 0, then for every short exact sequence of coherent

sheaves

0 → G^{∨} → E^{∨} ⟶^{q′} S → 0

admitting a morphism r : S → T of torsion sheaves with support in C_{smooth} for which q = r ∘ q′, h^{1}(C, G) equals 0.

Proof. (i) By Serre’s vanishing theorem, there exists an effective, ample divisor D in the smooth locus of C such that h^{1}(C, E(D)) equals 0. Define F = E(D), define E → F to be the obvious morphism E → E(D), and define T to be the cokernel of F^{∨} → E^{∨}.

(ii) This follows immediately from the semicontinuity theorem, cf. [Har77, §III.12].

(iii) There exists an injective morphism of coherent sheaves F → G with torsion cokernel. Because h^{1}(C, F) equals 0 and because h^{1} of every torsion sheaf is zero, the long exact sequence of cohomology implies that also h^{1}(C, G) equals 0. □

It is worth noting one interpretation of the sheaf F associated to a torsion quotient T . Assume that T is isomorphic to a direct sum of skyscraper sheaves at n distinct points c_{1}, …, c_{n} of C_{smooth}. (Inside the parameter space of torsion quotients, those with this property form a dense, open subset.) For each point c_{i}, the linear functional E^{∨}|_{c_{i}} ↠ T |_{c_{i}} gives a one-dimensional subspace Hom_{k}(T |_{c_{i}}, k) ↪ E|_{c_{i}}. The sheaf F is precisely the sheaf of rational sections of E having at worst a simple pole at each point c_{i} in the direction of this one-dimensional subspace of E|_{c_{i}}. This is often called an elementary transform up of E at the point c_{i} in the specified direction. So Lemma 3.3 says that h^{1} becomes zero after sufficiently many elementary transforms up at general points in general directions.

This interpretation is useful because the normal sheaf of a reducible curve can be understood in terms of elementary transforms up. To be precise, let Y be a k-scheme, let C be a proper, nodal curve, let C_{0} be a closed subcurve (i.e., a union of irreducible components of C), and let j : C → Y be a regular embedding such that Y is smooth at every node p_{1}, …, p_{n} of C which is contained in C_{0} and which is not a node of C_{0}. Then j_{0} : C_{0} → Y is also a regular embedding and both N_{C/Y} |_{C_{0}} and N_{C_{0}/Y} are locally free sheaves on C_{0}. For each i, there is a branch C_{i} of C at p_{i} other than C_{0}. Denote by T_{C_{i},p_{i}} the tangent direction of this branch in T_{Y,p_{i}}.

Lemma 3.4. [GHS03, Lemma 2.6] The restriction N_{C/Y} |_{C_{0}} equals the sheaf of rational sections of N_{C_{0}/Y} having at most a simple pole at each point p_{i} in the normal direction determined by T_{C_{i},p_{i}}.

Proof. The restrictions of N_{C/Y} |_{C_{0}} and N_{C_{0}/Y} to the complement of {p_{1}, …, p_{n}} are canonically isomorphic. The lemma states that this canonical isomorphism is the restriction of an injection N_{C_{0}/Y} ↪ N_{C/Y} |_{C_{0}} which identifies N_{C/Y} |_{C_{0}} with the sheaf of rational sections, etc. This local assertion can be verified in a formal neighborhood of each node p_{i}.

Locally near p_{i}, C → Y is formally isomorphic to the union of the two axes inside a 2-plane inside an n-plane, i.e., the subscheme of 𝔸_{k}^{n} with ideal I_{C/Y} = ⟨x_{1}x_{2}, x_{3}, …, x_{n}⟩. The branch C_{0} corresponds to just one of the axes, e.g., the subscheme of 𝔸_{k}^{n} with ideal I_{C_{0}/Y} = ⟨x_{2}, x_{3}, …, x_{n}⟩. The tangent direction of the other branch C_{i} is spanned by (0, 1, 0, …, 0). Thus it is clear that I_{C/Y} /I_{C_{0}/Y} ·I_{C/Y} is the submodule of I_{C_{0}/Y} /I_{C_{0}/Y}^{2} of elements whose fiber at 0 is contained in the annihilator of T_{C_{i},p_{i}}. Dualizing gives the lemma. □

The final bit of deformation theory needed has to do with deforming nodes. Let C be a proper, nodal curve and let j : C → Y be a regular embedding. Let p be a node of C and assume that Y is smooth at p. There are two branches C_{1} and C_{2} of C at p (possibly contained in the same irreducible component of C). The sheaf

T := Ext_{O_{C}}^{1} (Ω_{C}, O_{C})

is a skyscraper sheaf supported at p and with fiber canonically identified to

T |_{p} = T_{C_{1},p} ⊗_{k} T_{C_{2},p}.

The following lemma is as much definition as lemma.

Lemma 3.5. There exists a quotient of coherent sheaves

N_{C/Y} ↠ T

such that for both i = 1, 2 the quotient N_{C/Y} |_{C_{i}}/N_{C_{i}/Y} equals T . A first-order deformation of C ⊂ Y, i.e., a global section of N_{C/Y} is said to smooth the node p to first-order if the image of the section in T_{C_{1},p} ⊗_{k} T_{C_{2},p} is nonzero. For a deformation

C ⊂ Π ×_{k} Y

of C ⊂ Y over a smooth pointed curve (Π, 0) (i.e., C_{0} = C), if the associated first-order deformation of C ⊂ Y smooths the node p to first-order, then p is not contained in the closure of the singular locus of the projection,

(Π − {0}) ×_{Π} C → (Π − {0})

i.e., a general fiber C_{t} of the deformation smooths the node.

This is a well-known result. A good reference for this result, and many other results about deformations of singularities, is [Art76], particularly §I.6. Here is a brief remark on the proof. Because C ⊂ Y is a regular embedding, the conormal sequence is exact on the left, i.e.,

0 ⟶ I_{C/Y} /I_{C/Y}^{2} ⟶ Ω_{Y} |_{C} ⟶ Ω_{C} ⟶ 0

is a short exact sequence. Applying global Ext, there is a connecting map

δ : H^{0}(C, N_{C/Y}) → Ext_{O_{C}}^{1} (Ω_{C}, O_{C}).

There is also a local-to-global sequence for global Ext inducing a map

Ext_{O_{C}}^{1} (Ω_{C}, O_{C}) → H^{0}(C, Ext_{O_{C}}^{1} (Ω_{C}, O_{C})) = H^{0}(C, T ) = T_{C_{1},p} ⊗_{k} T_{C_{2},p}.

The composition of these two maps is precisely the map on global sections associated to N_{C/Y} → T . The global Ext group is identified with the first-order deformations of C as an abstract scheme, and the Ext term is identified with the first-order deformations of the node. It is worth noting that even if the first-order deformation does not smooth the node, the full deformation C ⊂ Π ×_{k} Y may smooth the node if the total space C is singular at (0, p).

The first result using the smoothing combs technique is the following.

Proposition 3.6. Let Y be a quasi-projective scheme over an algebraically closed field k. The very free locus Y_{v.f.} is an open subset of Y. More generally, for a flat, quasi-projective morphism π : Y → B, the relative very free locus Y_{π,v.f.} is an open subset of Y.

Let Y be an irreducible, quasi-projective scheme over an algebraically closed field k. Denote by t_{1}, resp. t_{2}, the closed point of ℙ_{k}^{1}, t_{1} = 0, resp. t_{2} = ∞. Let y_{1} and y_{2} be closed points of Y_{v.f.}, let a and k be nonnegative integers, and let there be given curvilinear k-jets in Y at each of y_{1} and y_{2}. If the given k-jets are general among all curvilinear k-jets at y_{1} and y_{2}, then there exists a morphism

f : (ℙ_{k}^{1}, t_{1}, t_{2}) → (Y_{v.f.}, y_{1}, y_{2})

mapping the k-jet of ℙ^{1} at t_{i} isomorphically to the given k-jet at y_{i} for i = 1, 2 and such that

f^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ a.

Proof. In the absolute case, resp. relative case, the very free locus Y_{v.f.}, resp. Y_{π,v.f.}, is defined to be the same as the very free locus of the smooth locus Y_{smooth}, resp. Y_{π,smooth}. Since the smooth locus is open in Y, and since an open subset of an open subset is an open subset, it suffices to prove the very free locus is open under the additional hypothesis that Y is smooth, resp. that π is smooth.

By the definition of Y_{v.f.}, for each i = 1, 2 there exists a very free morphism

f_{i} : (ℙ^{1}, 0) → (Y_{v.f.}, y_{i}), f_{i}^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ 1.

In particular, for each i = 1, 2, h^{1}(ℙ^{1}, f_{i}^{∗}T_{Y} (−\underline{0} − \underline{∞})) equals 0, where \underline{0}, resp. \underline{∞}, is the Cartier divisor of the point 0, resp. ∞, in ℙ^{1}. Since the normal sheaf of f_{i} is a quotient of f_{i}^{∗}T_{Y}, also h^{1}(ℙ^{1}, N_{f_{i}}(−\underline{0} − \underline{∞})) equals 0. Thus, applying Proposition 3.2 where Z_{1} = {0, ∞} and Z_{2} = ℙ^{1}, there exist deformations of the morphism f_{i} such that f_{i}(0) equals y_{i} and f_{i}(∞) is any point in a nonempty Zariski open subset of Y. The same argument holds in the relative case.

Next assume that Y is irreducible and quasi-projective. Then the smooth locus Y_{smooth} is also irreducible (or empty). Thus, by the same argument as above, a proof of the second result for smooth varieties implies the second result in general. Thus assume Y is also smooth.

Since Y is irreducible, the open for i = 1 intersects the open for i = 2. Thus there exist very free morphisms f_{1} and f_{2} such that f_{1}(∞) = f_{2}(∞). Let C be the nodal curve with two irreducible components C_{1} and C_{2} each isomorphic to ℙ^{1} and with a single node which, when considered as a point in either C_{1} or C_{2}, corresponds to ∞ in ℙ^{1}. Let f : C → Y be the unique morphism whose restriction to each component C_{i} equals f_{i}. Denote by

0 ⟶ N′_{C/Y} ⟶ N_{C/Y} ⟶ T ⟶ 0

the short exact sequence coming from Lemma 3.5. Using Lemma 3.4, there is an exact sequence

0 ⟶ N_{C/Y} |_{C_{1}}(−\underline{0} − \underline{∞}) ⟶ N′_{C/Y} (−y_{1} − y_{2}) ⟶ N_{C_{2}/Y} (−\underline{0}) ⟶ 0

and an exact sequence

0 ⟶ N_{C_{1}/Y} (−\underline{0} − \underline{∞}) ⟶ N_{C/Y} |_{C_{1}}(−\underline{0} − \underline{∞}) ⟶ κ_{∞} ⟶ 0

where κ_{∞} is the skyscraper sheaf on C_{1} supported at ∞. Applying the long exact sequence of cohomology, using that h^{1}(C_{i}, N_{C_{i}/Y} (−\underline{0} − \underline{∞})) equals 0 for i = 1, 2, and chasing diagrams, this finally gives that h^{1}(C, N′_{C/Y} (−y_{1} − y_{2})) also equals 0.

This has two consequences. First, this implies h^{1}(C, N_{C/Y} (−y_{1} − y_{2})) equals 0, and thus the space of deformations of C containing y_{1} and y_{2} is smooth by Proposition 3.2. And second, the map

H^{0}(C, N_{C/Y} (−y_{1} − y_{2})) → T_{C_{1},∞} ⊗ T_{C_{2},∞}

is surjective. Thus there exist first-order deformations of C containing y_{1} and y_{2} and smoothing the node at ∞. Since the space of deformations containing y_{1} and y_{2} is smooth, this first-order deformation is the one associated to a one-parameter deformation

C ⊂ Π ×_{k} Y

of [C] over a smooth, pointed curve (Π, 0) (e.g., choose Π to be a general complete intersection curve in the smooth deformation space containing the given Zariski tangent vector). By Lemma 3.5, for a general point t of Π, C_{t} is a smooth, connected curve containing y_{1} and y_{2}. Since the arithmetic genus of C is 0, the arithmetic genus of C_{t} is also 0, i.e., C_{t} ≅ ℙ_{k}^{1}. Let

f_{1} : ℙ_{k}^{1} → C_{t}

be an isomorphism with f_{1}(t_{i}) = y_{i} for i = 1, 2. Because h^{1}(C, N_{C/Y} (−y_{1} − y_{2})) equals 0, by the semicontinuity theorem also h^{1}(C_{t}, N_{C_{t}/Y} (−y_{1} −y_{2})) equals 0. This implies that

f_{1}^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), for integers a_{1}, …, a_{n} ≥ 1.

Next, for every integer a, let g_{a} : (ℙ^{1} → ℙ^{1} be the morphism z ↦ z^{a}. Then the composition f_{a} = f_{1} ∘ g_{1} is a morphism

f_{a} : (ℙ_{k}^{1}, t_{1}, t_{2}) → (Y_{v.f.}, y_{1}, y_{2})

with

f_{a}^{∗}T_{Y} = g_{a}^{∗}(f_{1}^{∗}T_{Y}) ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}) for integers a_{1}, …, a_{n} ≥ a,

namely the new integer a_{i}(f_{a}) equals a · a_{i}(f_{1}). Next, choosing a ≥ 2k + 1, this implies that

h^{1}(ℙ^{1}, f_{a}^{∗}T_{Y} (−(k + 1)(t_{1} + t_{2}))) = 0.

Applying Proposition 3.2 with ℙ^{1} ×_{k} Y in the place of Y, with the graph of f_{a} in the place of Z_{2} and with Z_{1} = (k + 1)(t_{1} + t_{2}) in the place of Z, deformations of f_{a} map the k-jet of ℙ^{1} at t_{1}, resp. at t_{2}, isomorphically to a general k-jet at y_{1}, resp. at y_{2}. □

The following proposition is the strongest generalization of Proposition 3.6 we will need. It is stated as a theorem about finding new sections of a rationally connected fibration under the hypothesis that one such section exists. In this sense it may seem premature (and dangerously close to circular logic), since Theorem 1.9 is not yet proved. In fact the proposition is used in the proof of Theorem 1.9 not for the original fibration, but only for a constant fibration

pr_{ℙ^{1}} : ℙ_{k}^{1} ×_{k} Y → ℙ_{k}^{1}

which obviously admits sections (constant sections). So there is nothing circular in the application of the proposition to the proof of Theorem 1.9.

Proposition 3.7 (Generic weak approximation). [KMM92], [HT06] Let B be a smooth, connected, projective curve over an algebraically closed field k. Let π : U → B be a smooth, quasi-projective morphism having irreducible geometric fibers. Assume there exists a section s : B → U mapping the generic point of B into the very free locus of the generic fiber of π. Let (b_{1}, …, b_{M}, b′_{1}, …, b′_{M′}) be distinct closed points of B such that s(b_{i}) is in the very free locus U_{b_{i},v.f.} of the fiber U_{b_{i}} for each i = 1, …, M. Let k and a be nonnegative integers. For each i, let x_{i} be a closed point of U_{b_{i},v.f.} and let there be given a curvilinear k-jet in U at x_{i}. Assuming each of these k-jets is a general k-jet at x_{i}, there exists a section σ : B → U such that

(i) for each i = 1, …, M, σ(b_{i}) equals x_{i},

(ii) for each i = 1, …, M′, σ(b′_{i}) equals s(b′_{i}),

(iii) for every invertible O_{B}-module L of degree ≤ a, h^{1}(B, N_{σ(B)/U} ⊗_{O_{B}} L^{∨})

equals 0, (iv) and σ maps the k-jet of b_{i} in B isomorphically to the given k-jet at x_{i} for

each i.

In fact Hassett and Tschinkel proved much more: the result holds for arbitrary k-jets transverse to the fibers of π (i.e., for k-jets whose associated Zariski tangent vector is not contained in a fiber of π). In what follows we only need the “generic” result, which is all we prove.

Proof. Denote by Ω_{π} the locally free sheaf of relative differentials of π, and denote by T_{π} the dual locally free sheaf. Choose a large integer N and enlarge the set of pairs ((b_{i}, x_{i}))|_{i=1,...,M} to a set ((b_{i}, x_{i}))_{i=1,...,N} having the same properties above and such that the collection (b_{i})_{i=M+1,...,N} is a general collection of N − M points in B (this is possible because for all but finitely many closed points of B, s(b) is contained in U_{b,v.f.}). By Proposition 3.6, applied with k = 1, i.e., in the case that k-jets are simply tangent directions, for every i = 1, …, N there exists a morphism

f_{i} : (ℙ^{1}, 0, ∞) → (U_{b_{i},v.f.}, s(b_{i}), x_{i})

such that

f_{i}^{∗}T_{π} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ 1

and the tangent direction of f_{i}(ℙ^{1}) at s(b_{i}) is a general tangent direction in T_{U_{b_{i}} ,s(b_{i})}. But of course the tangent space T_{U_{b_{i}} ,s(b_{i})} equals the normal space N_{s(B)/U}|_{s(b_{i})}. Thus the tangent direction of f_{i}(ℙ^{1}) at s(b_{i}) gives a general normal direction to s(B) in U at s(b_{i}).

Form the comb j : C_{comb} → U with handle s(B) and with each morphism f_{i} being a tooth L_{i} attached at s(b_{i}). By Lemma 3.4, N_{C_{comb}/U}|_{s(B)} equals the sheaf of rational sections of N_{s(B)/U} having at most a simple pole at each point s(b_{i}) in a general normal direction at s(b_{i}). Assuming the integer N is sufficiently large, Lemma 3.3 then implies that h^{1}(B, s^{∗}N_{C_{comb}/U}) equals 0. Moreover, fixing an auxiliary invertible sheaf M on B of degree g(B) + 1 and applying Lemma 3.3 to s^{∗}N_{s(B)/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M^{∨}, for N sufficiently large also h^{1}(B, s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M^{∨}) equals 0.

For every i, there is a short exact sequence

0 ⟶ f_{i}^{∗}N_{L_{i}/U_{b_{i}}} ⟶ f_{i}^{∗}N_{L_{i}/U} ⟶ f_{i}^{∗}N_{U_{b_{i}} /U} ⟶ 0.

Of course the normal sheaf N_{U_{b_{i}} /U} is just O_{U_{b_{i}}} since U_{b_{i}} is a smooth fiber of a morphism to a curve. Also the tangent direction of s(B) at s(b_{i}) surjects onto the fiber of N_{U_{b_{i}} /U} at s(b_{i}). Thus the elementary transform up of N_{L_{i}/U} at s(b_{i}) in this direction surjects onto the elementary transform up of O_{ℙ}^{1} at ∞, i.e., it surjects onto O_{ℙ^{1}}(1). Thus, by Lemma 3.4, there is a short exact sequence

0 ⟶ f_{i}^{∗}N_{L_{i}/U_{b_{i}}} ⟶ f_{i}^{∗}N_{C_{comb}/U} ⟶ O_{ℙ^{1}}(1) ⟶ 0.

Twisting by O_{ℙ^{1}}(−2) and applying the long exact sequence of cohomology associated to the short exact sequence, h^{1}(ℙ^{1}, f_{i}^{∗}N_{C_{comb}/U}(−\underline{0} − \underline{∞})) equals 0. Combined with the result of the previous paragraph and joining the two types of normal sheaf via the short exact sequence

0 → ⊕_{i=1}^{N}N_{C_{comb}/U}|_{L_{i}}(−x_{i} −s(b_{i})) → N_{C_{comb}/U}(−(x_{1} +⋯+x_{N})−(b′_{1} +⋯+b′_{M′}))

→ N_{C_{comb}/U}|_{s(B)}(−(b′_{1} + ⋯ + b′_{M′})) → 0,

the long exact sequence of cohomology implies both that

h^{1}(C_{comb}, N_{C_{comb}/U}(−(x_{1} + ⋯ + x_{N}) − (b′_{1} + ⋯ + b′_{M′}))) equals 0,

and that the map

H^{0}(C_{comb}, N_{C_{comb}/U}(−(x_{1}+⋯+x_{N})−(b′_{1}+⋯+b′_{M′}))) → H^{0}(B, s^{∗}N_{C_{comb}/U}(−(b′_{1}+⋯+b′_{M′})))

is surjective.

Thus, by Proposition 3.2, the space of deformations of C_{comb} containing x_{1}, …, x_{N} and b′_{1}, …, b′_{M′} is smooth. And, by Lemma 3.5, to prove there exists a deformation smoothing every node of C_{comb}, it suffices to prove for every i there exists a section of s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) whose image in T_{s(B),s(b_{i})} ⊗_{k} T_{L_{i},s(b_{i})} is nonzero. Of course this skyscraper sheaf T_{s(b_{i})} is a quotient of the fiber of s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) at b_{i}. Thus it suffices to prove for every i that h^{1}(B, s^{∗}N_{C_{comb}/U}(−b_{i} − (b′_{1} + ⋯ + b′_{M′}))) equals 0. Recall the auxiliary invertible sheaf M of degree g(B)+1. Because the invertible sheaf M(−b_{i}) has degree g(B), it is effective, say O_{B}(Δ_{i}). Thus there exists an injective O_{B}-module homomorphism

s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M^{∨} ↪

s^{∗}N_{C_{comb}/U}(−(b′_{1} + … b′_{M′})) ⊗_{O_{B}} M^{∨}(Δ_{i}) = s^{∗}N_{C_{comb}/U}(−b_{i} − (b′_{1} + ⋯ + b′_{M′}))

with torsion cokernel. Since h^{1}(B, s^{∗}N_{C_{comb}/U}(−(b′_{1} + ⋯ + b′_{M′})) ⊗_{O_{B}} M^{∨}) equals 0, and since every torsion sheaf has h^{1} equal to 0, also h^{1}(B, s^{∗}N_{C_{comb}/U}(−b_{i} − (b′_{1} + ⋯ + b′_{M′}))) equals 0 for every i. Therefore there exist a one-parameter family of deformations (C_{t})_{t∈Π} of C_{comb} containing each of x_{1}, …, x_{M}, containing each of s(b′_{1}), …, s(b′_{M′}) and smoothing every node of C_{comb}, i.e., for t general, C_{t} is smooth.

Because π_{U} maps s(B) to B with degree 1, also π_{U} maps C_{t} to B with degree 1. Because C_{t} is smooth, this means the projection C_{t} → B is an isomorphism. Therefore there exists a section σ_{t} : B → U of π_{U} with image C_{t}. In particular, σ_{t}(b_{i}) = x_{i} for every i = 1, …, M and σ_{t}(b′_{i}) = s(b′_{i}) for every i = 1, …, M′. Because h^{1}(C_{comb}, N_{C_{comb}/U}(−(x_{1} + ⋯ + x_{N}))) equals 0, by semicontinuity also h^{1}(B, σ_{t}^{∗}N_{σ_{t}(B)/U}(−(x_{1} + ⋯ + x_{N}))) equals 0 for t general. In particular, if N ≥ a + g(B), then for every invertible sheaf L of degree ≤ a, L^{∨}(x_{1} + ⋯ + x_{N}) has degree ≥ g(B) and thus is effective, say O_{B}(Δ). Therefore there exists an injective sheaf homomorphism

σ_{t}^{∗}N_{σ_{t}(B)/U}(−(x_{1}+⋯+x_{N})) ↪ σ_{t}^{∗}N_{σ_{t}(B)/U}(−(x_{1}+⋯+x_{N})+Δ) = σ_{t}^{∗}N_{σ_{t}(B)/U}⊗_{O_{B}}L^{∨}

with torsion cokernel. So, by the same type of argument as above, h^{1}(B, σ_{t}^{∗}N_{σ_{t}(B)/U}⊗_{O_{B}} L^{∨}) equals 0 for every invertible sheaf L of degree ≤ a.

Finally, applying the last result when a = (k + 1)(M + M′) and L = O_{B}((k + 1)(b_{1} + ⋯ + b_{M} + b′_{1} + ⋯ + b′_{M′})), there exists a section σ : B → U of π_{U} as above and satisfying h^{1}(B, σ^{∗}N_{σ(B)/U}(−(k + 1)(b_{1} + ⋯ + b_{M} + b′_{1} + ⋯ + b′_{M′}))) equals 0. Therefore, by Proposition 3.2 once more, for a general deformation of σ(B) containing x_{1}, …, x_{M} and s(b′_{1}), …, s(b′_{M′}), the k-jet of the curve at each point x_{i} and s(b′_{i}) is a general curvilinear k-jet in U at that point. □

The main application is to the case when U equals ℙ^{1}×_{k}Y where Y is a smooth, irreducible, quasi-projective k-scheme whose very free locus Y_{v.f.} is nonempty.

Corollary 3.8. Every rational curve in Y intersecting Y_{v.f.} is contained in Y_{v.f.}. For every integer k, for every integer a, for every collection of distinct, closed points b_{1}, …, b_{M} of ℙ^{1}, for every collection of closed points y_{1}, …, y_{M} of Y_{v.f.} (not necessarily distinct), and for every choice of a curvilinear k-jet in Y at each point y_{i}, if each k-jet is general among curvilinear k-jets at y_{i}, then there exists a morphism

f : (ℙ^{1}, b_{1}, …, b_{M}) → (Y, y_{1}, …, y_{M})

mapping the k-jet of ℙ^{1} at b_{i} isomorphically onto the given k-jet at y_{i} and such that

f^{∗}T_{Y} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}), a_{1}, …, a_{n} ≥ a.

Proof. Let B = ℙ^{1}, let U = B×_{k}Y and let π_{B} be the obvious projection. The sections of π_{B} are precisely the graphs of morphisms f : ℙ^{1} → Y. In particular, if f is a morphism whose image intersects Y_{v.f.}, then the section s = (Id_{B}, f) satisfies the hypotheses of Proposition 3.7. Thus, for every point b′ = b′_{1} of ℙ^{1}, there exists a section σ = (Id_{ℙ^{1}}, ϕ) with σ(b′) = s(b′) and with h^{1}(B, σ^{∗}N_{σ(B)/U}(−2)) equal to 0. In other words, ϕ : ℙ_{k}^{1} → Y is a morphism with ϕ(b′) = f(b′) and with h^{1}(ℙ^{1}, ϕ^{∗}T_{Y} (−2)) equal to 0. Thus ϕ is a very free morphism whose image contains f(b′). Therefore every point in the image of f is contained in the very free locus, i.e., every rational curve in Y intersecting Y_{v.f.} is contained in Y_{v.f.}.

The rest of the corollary is just a straightforward translation of Proposition 3.7 to this context. □

There is one more result in this direction which is useful. The proof is similar to the arguments above.

Lemma 3.9. [Kol96, Lemma II.7.10.1] Let C_{comb} be a comb with handle C and teeth L_{1}, …, L_{n}. Let ρ : C → Π be a one-parameter deformation of C_{comb} over a pointed curve (Π, 0) whose general fiber C_{t} is smooth. Let E be a locally free sheaf on C. If E|_{L_{i}} is ample for every i and if h^{1}(C, (E|_{C}) ⊗_{O_{C}} M) equals 0 for every invertible O_{C}-module M of degree ≥ n, then h^{1}(C_{t}, E|_{C_{t}}) equals 0 for general t in Π.

4. Ramification issues

The argument sketched in Section 2 and the powerful smoothing combs technique from Section 3 form the core of the proof of Theorem 1.9. However there is a technical issue complicating matters. There may be codimension 1 points of X at which the morphism π : X → B is not smooth. In other words, finitely many scheme-theoretic fibers of π may have irreducible components occurring with multiplicity ≥ 1. This is a well-known issue when working with fibrations. Although there are sophisticated ways to deal with this (using log structures or Deligne-Mumford stacks), for the purposes of this proof it suffices to deal with this in a more naive manner.

In fact there may be codimension 0 points of X at which π is not smooth, at least if k has positive characteristic. The hypotheses in Theorem 1.9 prevent this, but something slightly weaker suffices. Let B be a smooth k-curve, let X be a reduced, finite type k-scheme and let π : X → B be a flat morphism. From here on, we assume the following hypothesis.

Hypothesis 4.1. The geometric generic fiber of π is reduced. Equivalently, π is smooth at every generic point of X, cf. [Gro67, Proposition 4.6.1]. This hypothesis is automatic if char(k) equals 0.

Definition 4.2. The good locus of π is the maximal open subscheme U of X such that U is smooth and such that for every point b of B the reduced scheme of the fiber π^{−1}(b) ∩ U is smooth. Denote the restriction of π to U by π_{U}. The morphism π is good if the good locus equals all of X. The log divisor of π is the Cartier divisor D_{π,log} of U given by

D_{π,log} := ∑_{b∈B(k)} π_{U}^{∗}(b) − π_{U}^{∗}(b)_{red},

where π_{U}^{∗}(b)_{red} is the reduced Cartier divisor.

Since the geometric generic fiber of π is reduced, so is the geometric generic fiber of π_{U} (or else it is empty if U is empty). Thus the sum in the definition of the log divisor reduces to a sum over those finitely many closed points b of B for which π_{U}^{∗}(b) is nonreduced.

Lemma 4.3. The complement of U in X has codimension ≥ 2. If char(k) equals 0, then the pullback map on relative differentials

π_{U}^{∗} : π_{U}^{∗}Ω_{B/k} → Ω_{U/k}

factors uniquely through the inclusion

π_{U}^{∗}Ω_{B/k} ↪ π_{U}^{∗}Ω_{B/k}(D_{π,log})

and the cokernel

Ω_{π,log} := Coker(π_{U}^{∗}Ω_{B/k}(D_{π,log}) → Ω_{U/k})

is locally free.

Proof. To construct U, first remove the closure of the singular locus of the geometric generic fiber of π and next remove the singular locus from the reduced scheme of the finitely many singular fibers. Both of these sets have codimension 2 in X (the first by Hypothesis 4.1).

The proof of the second part uses that char(k) = 0. It can be checked formally locally near every closed point x of U. Denote by b the image π(x) in B and denote by D the reduced structure on the irreducible component of π^{−1}(b) containing x. Since x is in U, D is a smooth Cartier divisor in U. Let r be a defining equation for D in U and let t be a defining equation for b in B. Near x, π^{∗}(b) = mD+ other terms. Thus, in Ô_{U,x},

π^{∗}t = a_{m}r^{m} + a_{m+1}r^{m+1} + …

where a_{m} is a unit. Because char(k) = 0, the power series

u = \sqrt[m]{a_{m} + a_{m+1}r + …}

is a well-defined unit in Ô_{U,x}. Thus, after replacing r by ur, there exists a regular system of parameters r, r_{2}, …, r_{n} for Ô_{U,x} with respect to which the pullback homomorphism π^{∗} is the unique local homomorphism with

π^{∗}t = r^{m} and π^{∗}(dt) = mr^{m−1}dr.

In particular, the pullback homomorphism π^{∗} on relative differentials locally factors through π^{∗}Ω_{B}((e−1)D) = π^{∗}Ω_{B}(D_{π,log}). Moreover the stalk of the cokernel of the induced homomorphism is the free module generated by dr_{2}, …, dr_{n}. □

The locally free quotient Ω_{π,log} of Ω_{π} is called the sheaf of log relative differentials. Of course it equals the torsion-free quotient of Ω_{π}. But its true importance comes from the following lemma: given a base change V → B for which the normalized fiber product { \atop U ×_{B} V} is smooth over V, the sheaf Ω_{{ \atop U×_{B}V} /V} of relative

differentials of { \atop U ×_{B} V} over V equals the pullback of Ω_{π,log}. Thus the relative deformation theory of { \atop U ×_{B} V} over V is already captured by the sheaf Ω_{π,log} on U. Before stating the lemma precisely, there is some setup.

Let

π : U → B, ϖ : V → B

be two good morphisms with respective log divisors D_{π,log} and E_{ϖ,log}. Let b be a closed point of B. Let D be a prime divisor of U in Supp(D_{π,log}) ∩ π^{−1}(b), and let E be a prime divisor of V in Supp(E_{ϖ,log}) ∩ ϖ^{−1}(b). Denote by m_{D} − 1, resp. m_{E} − 1, the coefficient of D in D_{π,log}, resp. the coefficient of E in E_{ϖ,log}. The normalized fiber product of U and V along D and E is the normalization { \atop U ×_{B} V} of U ×_{B} V along D ×_{{b}} E. Denote by

pr_{U} : U ×_{B} V → U, pr_{V} : U ×_{B} V → V

the two projections, and denote by

p̃r_{U} : { \atop U ×_{B} V} → U, p̃r_{V} : { \atop U ×_{B} V} → V

the compositions with the normalization morphism. Denote by Exc the exceptional locus of the morphism, i.e.,

Exc := (p̃r_{U}^{−1}(D) ∩ p̃r_{V}^{−1}(E))_{reduced}.

From this point forward we explicitly assume that char(k) equals 0.

Hypothesis 4.4. The algebraically closed ground field k has characteristic 0. In particular, this implies Hypothesis 4.1.

The sheaves Ω_{π} and Ω_{π,log} agree on a dense open subset of U, namely U − Supp(D_{π,log}). Because p̃r_{V} and pr_{V} are isomorphic over a dense open subset of V (namely V − E) also Ω_{p̃r_{V}} agrees with p̃r_{V}^{∗} Ω_{π} on a dense open subset of { \atop U ×_{B} V}.

Therefore also Ω_{ϖ} agrees with p̃r_{V}^{∗} Ω_{π,log} on a dense open subset of { \atop U ×_{B} V}.

Lemma 4.5. The morphism

p̃r_{V} : { \atop U ×_{B} V} → V

is smooth at every point of Exc if and only if m_{D} divides m_{E}. In this case the reduced normalization equals the blowing up of U ×_{B} V along the closed subscheme pr_{U}^{−1}(D) × pr_{V}^{−1}((m_{E}/m_{D})E) and Exc is contained in the maximal open neighborhood of { \atop U ×_{B} V} on which Ωπ̃ agrees with (p̃r_{V})^{∗}Ω_{π,log}.

Proof. This is proved in much the same way as the second part of Lemma 4.3. For every closed point x of U and y of V with common image point b = π(x) = ϖ(y), there exist a regular system of parameters (r, r_{2}, …, r_{n}) for Ô_{U,x}, resp. (s, s_{2}, …, s_{p}) for Ô_{V,y}, and a regular parameter t for Ô_{B,b} such that

π^{∗}t = r^{m_{D}} and ϖ^{∗}t = s^{m_{E}},

and thus,

Ô_{U×_{B}V,(x,y)} = k \begin{aligned}[ \\ [ & r, r_{2}, …, r_{n}, s, s_{2}, …, s_{p} & ] \\ & ] \end{aligned} /⟨r^{m_{D}} − s^{m_{E}}⟩.

Denoting by m the greatest common factor of m_{D} and m_{E}, the stalk of the normalization equals

k \begin{aligned}[ \\ [ & u, r, r_{2}, …, r_{n}, s, s_{2}, …, s_{p} & ] \\ & ] \end{aligned} /⟨r − u^{m_{E}/m}, s − u^{m_{D}/m}⟩.

Thus it is formally smooth as a k \begin{aligned}[ \\ [ & s, s_{2}, …, s_{p} & ] \\ & ] \end{aligned}-algebra if and only if m_{D}/m equals 1, i.e., if and only if m_{D} divides m_{E}. In this case it is easy to see that the normalization is the blowing up at the ideal ⟨s, t^{m_{E}/m_{D}}⟩ and it is easy to see that the module of relative differentials is the free module generated by dr_{2}, …, dr_{n}, i.e., it is the pullback of Ω_{π,log}. □

Having introduced the ideas need to deal with the ramification issues, we now resume the proof of Theorem 1.9. So from this point on we assume the following.

Hypothesis 4.6. The following hypotheses of Theorem 1.9 hold.

(i) The algebraically closed ground field k has characteristic 0, i.e., Hypothesis 4.4 holds.

(ii) The smooth k-curve B is projective and connected.

(iii) The reduced, finite type k-scheme X is normal and projective.

(iv) And the geometric generic fiber of the flat morphism π : X → B is a normal, integral scheme whose smooth locus contains a very free curve.

Definition 4.7. A log preflexible curve is a connected, smooth, proper curve C ⊂ U such that

(i) the generic fiber of C over B is contained in the very free locus of the generic fiber of U over B,

(ii) π_{U}(C) equals B, Figure 4. Where a log preflexible curve intersects a reduced, possibly singular, fiber the map from the curve to B is unramified.

(iii) and every intersection point of C with supp(D_{π,log}) is transverse, i.e., the tangent direction of C at the intersection point is not contained in the tangent space of supp(D_{π,log}).

A linked log preflexible curve is a B-morphism from a linked curve j : C_{link} → U such that the handle C is log preflexible and for every link L_{i} the image in B of L_{i} is disjoint from the image in B of D_{π,log}.

A log preflexible curve C is a log flexible curve if

h^{1}(C, T_{π,log}|_{C}) equals 0, where T_{π,log} := Hom_{O_{U}} (Ω_{π,log}, O_{U}).

A linked log preflexible curve is a linked log flexible curve if

h^{1}(C_{link}, j^{∗}T_{π,log}) equals 0.

Figure 4 shows a log preflexible curve intersecting a singular, but reduced fiber. Because the curve is transverse to the fiber, the morphism to B is unramified. On the other hand, Figure 5 shows a log preflexible curve intersecting a nonreduced fiber – the middle component has multiplicity 2. Necessarily the map from the curve to B is ramified.

Lemma 4.8. There exists a log preflexible curve C. In fact, every intersection of X with dim(X) − 1 general hyperplanes is a log preflexible curve.

Proof. Because X−U has codimension 2 in X, a general complete intersection curve in X is disjoint from X −U, i.e., it is contained in U. By hypothesis, U_{π,v.f.} is Figure 5. Where a log preflexible curve intersects a nonreduced fiber – e.g., the middle component has multiplicity 2 – the map from the curve to B is necessarily ramified.

a dense open subset of U and thus a general complete intersection curve intersects this open. Finally, by Bertini’s theorem a general complete intersection curve in U is smooth and intersects supp(D_{π,log}) transversally. □

An important consequence of the smoothing combs technique is the following result.

Proposition 4.9. There exists a log flexible curve in X. In fact, for every comb in X with log preflexible handle C and with sufficiently many very free teeth in fibers of π_{U} attached at general points of C and with general tangent directions, there exists a one-parameter deformation of the comb whose general member is a log flexible curve.

Proof. By hypothesis, C intersects the very free locus U_{π,v.f.} of the morphism π_{U}. By the same argument as in the proof of Proposition 3.6, U_{π,v.f.} is open. Therefore all but finitely many points of C are contained in U_{π,v.f.}. By Proposition 3.6 applied to 1-jets, i.e., to tangent directions, for each such point c there exists a very free rational curve in U_{π_{U} (c)} containing c and whose tangent direction at c is a general tangent direction in U_{π_{U} (c)}.

Let C_{comb} be a comb obtained by attaching to C a number of teeth L_{1}, …, L_{N} as in the previous paragraph at general points of C (in particular, points where C → B is unramified) and with general tangent directions in U_{π_{U} (c)}. These tangent directions are the same as normal directions to C in U. By the same argument as in the proof of Proposition 3.7, if N is sufficiently large there is a one-parameter deformation

C ⊂ Π ×_{k} U

of C_{comb} such that C_{t} is smooth for general t in Π. The properties (i), (ii) and (iii) of Definition 4.7 are all open properties and hold for C_{0} = C_{comb}, thus also hold for C_{t} so long as t is general.

For each tooth L_{i} in a fiber U_{b_{i}}, T_{π,log}|_{L_{i}} equals T_{U_{b_{i}}} |_{L_{i}}. Since L_{i} is very free, this is an ample locally free sheaf. Thus, by Lemma 3.9 with the pullback of T_{π,log} in the place of E, we have that h^{1}(C_{t}, T_{π,log}|_{C_{t}}) equals 0 for t a general point of Π. Therefore, for t a general point of Π, C_{t} is a log flexible curve. □

Because the fibers of π are rationally connected, every log preflexible curve, resp. log flexible curve, extends to a linked log preflexible curve, resp. linked log flexible curve.

Lemma 4.10. For every linked curve C_{link} such that each point b_{i} = π_{C,link}(L_{i}) is disjoint from π_{U}(D_{π,log}), and for every B-morphism j_{0} : C → X mapping C isomorphically to a log preflexible curve, resp. log flexible curve, and mapping each fiber C_{b_{i}} into the very free locus U_{π,v.f.} of π_{U}, there exists a B-morphism j : C_{link} → X which is linked log preflexible, resp. linked log flexible, and restricting to j_{0} on C.

Proof. Let L_{i} be a link of C_{link}. Let L_{i} intersect C in m points t_{1}, …, t_{m} contained in the fiber over a general point b_{i} of B. Let x_{1}, …, x_{m} be the images j(t_{1}), …, j(t_{m}) in U_{b_{i},v.f.}. By Corollary 3.8, there exists a morphism

j_{i} : (L_{i}, t_{1}, …, t_{m}) → ((U_{b_{i},v.f.}, x_{1}, …, x_{m})

such that

j_{i}^{∗}T_{U_{b_{i}}} ≅ O_{ℙ^{1}}(a_{1}) ⊕ ⋯ ⊕ O_{ℙ^{1}}(a_{n}) for integers a_{1}, …, a_{n} ≥ m − 1.

Because of this,

h^{1}(L_{i}, j_{i}^{∗}T_{U_{b_{i}}} (−(t_{1} + ⋯ + t_{m}))) equals 0.

Define j : C_{link} → U to be the unique morphism restricting to j_{0} on C and restricting to j_{i} on each link L_{i}. Because j_{i}(t_{k}) = j_{0}(t_{k}) for every link L_{i} and for every node t_{k} contained in L_{i}, this morphism is defined. It is clearly log preflexible.

Next assume that j_{0} is log flexible. The claim is that j is also log flexible. To see this, consider the short exact sequence

0 ⟶ ⊕_{i}j_{i}^{∗}T_{U_{b_{i}}} (−C_{b_{i}}) ⟶ j^{∗}T_{π,log} ⟶ j_{0}^{∗}T_{π,log} ⟶ 0.

By the hypothesis that j_{0} is log flexible, the third term has vanishing h^{1}. And by the construction of j_{i}, j_{i}^{∗}T_{U_{b_{i}}} (−C_{b_{i}}), i.e., j_{i}^{∗}T_{U_{b_{i}}} (−(t_{1} + ⋯ + t_{m})), has vanishing h^{1}. Thus, by the long exact sequence of cohomology, also h^{1}(C_{link}, j^{∗}T_{π,log}) equals 0. Therefore j : C_{link} → U is a linked log flexible curve. □

5. Existence of log deformations

There is a definition of one-parameter deformation that takes the divisor D_{π,log} into account. Unfortunately, not every curve over B admits a log deformation specializing to a section curve, e.g., étale covers of B are rigid. However, after attaching a sufficient number of links, the linked curve does admit a log deformation specializing to a section curve.

Definition 5.1. Let (C_{link}, π_{C,link}) be a linked curve with handle C. Let D_{C} ⊂ C be an effective, reduced, Cartier divisor contained in the smooth locus of C_{link}. A one-parameter log deformation of (C_{link}, π_{C,link}, D_{C}) is a one-parameter deformation of (C_{link}, π_{C,link}),

(ρ, π_{C}) : C → Π ×_{k} B

together with an effective Cartier divisor D_{C} ⊂ C such that (i) the pullback of D_{C} to C_{0} = C_{link} equals D_{C} (ii) and π_{C}(D_{C}) equals π_{C}(D_{C}), i.e., D_{C} is vertical over B.

Lemma 5.2. For every finite morphism of smooth, projective curves π_{C} : C → B and for every effective, reduced, Cartier divisor D_{C} of C, after attaching sufficiently many links to C over general points of B, there exists a one-parameter log deformation specializing to a section curve.

Proof. For all sufficiently positive integers e, for a general morphism g : C → ℙ^{1} of degree e, the induced morphism (π_{C}, g) : C → B ×_{k} ℙ^{1} is unramified and is injective except for finitely many double points, none of which intersects the image of D_{C}. Denote by Σ → B ×_{k} ℙ^{1} the blowing up along the finitely many double points of (π_{C}, g)(C). Then there is a B-morphism h : C → Σ which is an embedding.

For each point p of D_{C}, denote by m_{p} the multiplicity of p in the Cartier divisor π_{C}^{∗}(π_{C}(p)). Denote by ν_{p} : Σ′_{p} → Σ the m_{p}-fold iterated blowup of Σ first at p, then at the image of p in the strict transform of h(C), etc. Denote by E_{p} the final exceptional divisor of this sequence of blowups. The point of this construction is that the strict transform of h(C) intersects E_{p} at p, and E_{p} occurs with multiplicity m_{p} in the Cartier divisor Σ′_{p} ×_{B} {π_{C}(p)}. Denote by ν : Σ′ → Σ the fiber product over all points p in D_{C} of ν_{p} : Σ′_{p} → Σ. Denote by E the Cartier divisor in Σ′ being the sum over all p of the pullback of E_{p} from Σ′_{p}. Denote by π_{Σ′} : Σ′ → B the composition of Σ′ → Σ → B ×_{k} ℙ^{1} with pr_{B}. Denote by h′ : C → Σ′ the strict transform of h(C). The point of this construction is that E is a Cartier divisor in Σ′ which is vertical over B and such that h^{∗}E equals D_{C}.

Denote by d the degree of π_{C} and let t_{1}, …, t_{d} be closed points of ℙ^{1} such that the Cartier divisor B ×_{k} {t_{1}, …, t_{d}} of B ×_{k} ℙ^{1} is disjoint from all double points of (π_{C}, g)(C) and disjoint from (π_{C}, g)(D). Denote by T the strict transform of B ×_{k} {t_{1}, …, t_{d}} in Σ′. Form the invertible sheaf O_{Σ′}(h′(C) − T) and the pushforward E := π_{Σ′,∗}O_{Σ′}(h′(C) − T) on B. Because π_{Σ′} is flat and because O_{Σ′}(h′(C) − T) is locally free, E is torsion-free. For every point b in B − π_{C}(D_{C}), Σ′_{b} is isomorphic to ℙ^{1} (via the projection Σ′ → B ×_{k} ℙ^{1} → ℙ^{1}). And Σ′_{b} ∩ h′(C) and Σ′_{b} ∩ T are divisors of the same degree d. Thus O_{Σ′}(h′(C)−T)|_{Σ′_{b}} is isomorphic to O_{Σ′_{b}} ≅ O_{ℙ^{1}}. Therefore E|_{b} is isomorphic to H^{0}(Σ′_{b}, O_{Σ′_{b}}), which is one-dimensional. Therefore E is an invertible sheaf.

By Riemann-Roch and Serre duality, for every sufficiently large degree, for a general effective divisor Δ on B of that degree, E ⊗_{O_{B}} O_{B}(Δ) is globally generated. Choose Δ to be disjoint from π_{C}(D_{C}) and from the image in B of the finitely many intersection points of h′(C) and T. Since E ⊗_{O_{B}} O_{B}(Δ) is globally generated, there exists a section which is nonzero at every point of Δ. Of course a nonzero section of this sheaf (up to scaling) is precisely the same thing as a divisor V on Σ′ such that

h′(C) + π_{Σ′}^{∗}Δ ∼ T + V.

For b in B − π_{C}(D_{C}), if the section is nonzero at b then V does not intersect Σ′_{b}. The same does not necessarily hold for points b of π_{C}(D_{C}) since b may lie in the support of R^{1}π_{Σ′,∗}O_{Σ′}(h′(C)−T). Therefore V is a sum of finitely many irreducible components of fibers of π_{Σ′} (possibly with multiplicity) lying over points not in Δ.

The linked curve (C_{link}, π_{C,link}) is h′(C)+π_{Σ′}^{∗}Δ together with the restriction of π_{Σ′}. Denote by Π the pencil of divisors in Σ′ spanned by the divisors h′(C) + π_{Σ′}^{∗}Δ and T + V, with these two divisors marked as 0 and ∞ respectively. Denote by C ⊂ Π×_{k} Σ′ the corresponding family of divisors. By Bertini’s theorem, the general member C_{t} is smooth away from the base locus. Now the only singular points of h′(C) + π_{Σ′}^{∗}Δ are the points h′(π_{C}^{−1}(Δ)). Since V does not intersect π_{Σ′}^{∗}Δ, these singular points are not in the base locus. Since C_{0} is nonsingular at every basepoint, the same is true for C_{t} for t general. Thus a general member C_{t} is smooth everywhere.

Define D_{C} to be the pullback to C of the Cartier divisor E in Σ′. Because E is vertical over B and because h^{∗}E equals D_{C}, the deformation C together with the effective Cartier divisor D_{C} is a one-parameter log deformation of (C_{link}, π_{C,link}, D_{C}). And it specializes at t = ∞ to a union of section curves and vertical curves. □

6. Completion of the proof

We are finally prepared for the proof of Theorem 1.9.

Proof of Theorem 1.9. By Proposition 4.9, there exists a log flexible curve j_{0} : C → U. Denote by D_{C} the reduced scheme of the intersection C ∩ D_{π,log}. By Lemma 5.2, after attaching finitely many links to C over the points of a general divisor Δ of B, the linked curve C_{link} together with D_{C} admits a one-parameter log deformation

(ρ, π_{C}) : C → Π ×_{k} B, D_{C} ⊂ C

of (C_{link}, D_{C}) specializing to a section curve (in fact C_{∞} is a union of section curves and vertical curves).

By Proposition 3.6, the relative very free locus U_{π,v.f.} is open in U. Thus C ∩ U_{π,v.f.} is open in C. So its complement is finitely many points in C. Thus a general divisor Δ is disjoint from the finite set π_{U}(D_{π,log}) and from the finite set π_{C}(C − C ∩ U_{π,v.f.}). Then, by Lemma 4.10, there exists an extension of j_{0} to a linked log flexible curve

j : C_{link} → U.

Form the fiber product

U_{C} := C ×_{π_{C},B,π_{U}} U.

Since π_{U} is flat, also the projection

pr_{C} : U_{C} → C

is flat. Since π_{C} is surjective, the geometric generic fiber of pr_{C} equals the geometric generic fiber of π_{U}, which is integral. Since pr_{C} is flat with integral geometric generic fiber, U_{C} is integral. Define

ν : Ũ_{C} → U_{C}

to be the blowing up of U_{C} along the closed subscheme D_{C} ×_{B} D_{π,log}. Since U_{C} is integral, also Ũ_{C} is integral. And the composition

Ũ_{C} → U_{C} → C → Π

is surjective. Since Π is a smooth curve, the morphism

ρ̃: Ũ_{C} → Π

is flat.

Consider the graph,

Γ_{j} : C_{link} = C_{0} → C_{0} ×_{B} U = U_{C,0}.

Because the links of C_{link} do not intersect D_{π,log}, the image of Γ_{j} is smooth at every point of intersection with D_{C} ×_{B} D_{π,log}. Since ν is birational, Γ_{j} gives a rational transformation from C_{link} to Ũ_{C,0}. Since ν is proper, and since C_{link} is smooth at every point of intersection with D_{C} ×_{B} D_{π,log}, the valuative criterion of properness implies this rational transformation is actually a regular morphism

Γ̃_{j} : C_{0} → Ũ_{C,0}.

Clearly this is a section of the projection morphism

pr_{C_{0}} : Ũ_{C,0} → C_{0}.

For every point t in C_{link} − D_{C}, the morphism π_{U} : U → B is smooth at j(t). Therefore also U_{C,0} → C_{0} is smooth at Γ_{j}(t). And since ν is an isomorphism over Γ_{j}(t), also pr_{C_{0}} : Ũ_{C,0} → C_{0} is smooth at Γ̃_{j}(t). Also the vertical tangent bundle equals the pullback of the vertical tangent bundle of π_{U} : U → B, which also equals T_{π,log} (since j(t) is not in D_{π,log}).

Let t be a point of D_{C} and let D_{t} be the unique irreducible component of D_{π,log} containing j(t). Give D_{t} the reduced structure. Because j_{0}(C) is transverse to D_{t} at j_{0}(t), the ramification index m_{C} − 1 of π_{C} : C → B at t equals the ramification index m_{D} − 1 of π_{U} along D_{t}. Therefore, by Lemma 4.5, the projection

pr_{C_{0}} : Ũ_{C,0} → C_{0}

is smooth over the preimage of {t} × D_{t} for every t and the vertical tangent bundle equals the pullback of T_{π,log}. Since Γ_{j}(t) is in {t} × D_{t}, this implies that pr_{C_{0}} is smooth at every point of the image of Γ̃_{j} and the vertical tangent bundle of pr_{C_{0}} equals the pullback of T_{π,log}.

Since Γ̃_{j} is a section with image in the smooth locus of pr_{C_{0}}, the normal sheaf N equals the restriction of the vertical tangent bundle. Therefore Γ̃_{j}^{∗}N equals j^{∗}T_{π,log}. Since j : C_{link} → U is log flexible, h^{1}(C_{link}, j^{∗}T_{π,log}) equals 0. Therefore, by Proposition 3.1, the relative Hilbert scheme Hilb(Ũ_{C}/Π) is smooth over Π at the point 0′ := [Image(Γ̃_{j})]. Thus for a general complete intersection curve Π′ containing 0′, the morphism Π′ → Π is smooth at 0′.

Replace Π′ by the unique irreducible component containing 0′, and then replace this by its normalization. The result is that Π′ is a smooth, projective, connected curve together with a morphism Π′ → Hilb(Ũ_{C}/Π) so that the induced morphism Π′ → Π is smooth at 0′. In particular it is flat, so surjective. Let ∞′ denote a closed point of Π′ mapping to ∞. Then (Π′, 0′, ∞′) → (Π, 0, ∞) is a flat morphism of 2-pointed smooth curves. Thus, by Lemma 2.2, the base change Π′ ×_{Π} C is a one-parameter deformation of C_{link} over (Π′, 0′, ∞′) specializing to a section curve.

Denote by

Z ⊂ Π′ ×_{Π} Ũ_{C}

the pullback of the universal closed subscheme Univ(Ũ_{C}/Π) by the morphism Π′ → Hilb(Ũ_{C}/Π). The composition with pr_{C} is a projective morphism

Z ⊂ Π′ ×_{Π} Ũ_{C} → Π′ ×_{Π} C

of flat Π′-schemes. Moreover, the fiber over 0′ ∈ Π′ is an isomorphism since the projection Γ̃_{j}(C_{link}) → C_{link} is an isomorphism. Therefore the morphism is an isomorphism over N ×_{Π} C for some open neighborhood N of 0′ in Π′. (This is wellknown; a complete proof is given in [dJS03, Lemma 4.7].) Invert this isomorphism and compose it with the morphism

Π′ ×_{Π} Ũ_{C} → Ũ_{C} → U_{C} → U.

The result is precisely an extension

j_{N} : N ×_{Π} C → X

of j for the one-parameter deformation Π′ ×_{Π} C. Therefore, by Lemma 2.3, there exists a section s : B → X of π. □

7. Corollaries

There are a number of consequences of Theorem 1.9 and its generalization to positive characteristic in [dJS03]. Many of these consequences were recognized before Conjecture 1.8 was proved.

Corollary 7.1. [Kol96, Conjecture IV.5.6] Conjecture 1.7 is true. Moreover, for every smooth, projective, irreducible variety X over an algebraically closed field of characteristic 0, there exists a dense open X^{0} ⊂ X and a projective, smooth morphism q_{0} : X^{0} → Q^{0} such that every fiber of q_{0} is rationally connected, and every projective closure of Q^{0} is nonuniruled.

Corollary 7.2. [GHS03, Corollary 1.7] The uniruledness conjecture implies Mumford’s conjecture. To be precise, assume that for every smooth, projective, irreducible variety X over an algebraically closed field k of characteristic 0, if X is nonuniruled then h^{0}(X, ω_{X}^{⊗n}) is nonzero for some n > 0. Then for every smooth, projective, irreducible variety X over k, if X is not rationally connected then h^{0}(X, Ω_{X}^{⊗n}) is nonzero for some n > 0.

The next corollary is a fixed point theorem. In characteristic 0 it can be proved using the Atiyah-Bott fixed point theorem. But in positive characteristic it is a new result. There are examples due to Shioda proving one cannot replace “separably rationally connected” by “rationally connected”, cf. [Shi74].

Corollary 7.3. [Kol03] Let Y be a smooth, projective, separably rationally connected variety over a field k and let f : Y → Y be a k-automorphism. If char(k) is positive, say p, assume in addition that f has finite order n not divisible by p^{2}. Then the fixed locus of f is nonempty.

Proof. Of course it suffices to prove the case when k is algebraically closed, since the fixed locus of the base change equals the base change of the fixed locus. First assume f has finite order n. If n is prime to char(k), let B′ denote ℙ^{1} and let ℤ/nℤ act on ℙ^{1} by multiplication by a primitive n^{th} root of unity. Note this action fixes ∞ and has trivial generic stabilizer. If char(k) = p is positive and if n = pm where m is prime to p, let B′ be the normal, projective completion of the affine curve

𝕍(y^{m} − (x^{p} − x)) ⊂ 𝔸_{k}^{2}.

Let ζ be a primitive m^{th} root of unity, and let a generator of ℤ/mℤ act by (x, y) ↦ (x, ζy). Similarly, let a generator of ℤ/pℤ act by (x, y) ↦ (x + 1, y). Clearly these actions commute, thus define an action of ℤ/nℤ on B′. Note this action fixes the unique point ∞ not in the affine chart above, and the action has trivial generic stabilizer.

Let ℤ/nℤ act diagonally on Y ×_{k} B′, and let X be the quotient. Also let B be the quotient of the ℤ/nℤ-action on B′. The projection π : X → B satisfies the hypotheses of Theorem 1.9 (or its generalization in [dJS03]). Therefore there exists a section. This is the same as ℤ/nℤ-equivariant k-morphism f : B′ → Y. In particular, since ∞ is a fixed point in B′, f(∞) is a fixed point in Y.

Next assume k has characteristic 0. By general limit arguments there exists an integral, finitely generated ℤ-algebra R, a ring homomorphism R ↪ k, a smooth, projective morphism Y_{R} → Spec R whose relative very free locus is all of Y_{R}, and an R-automorphism f_{R} : Y_{R} → Y_{R} such that the base change Y_{R} ⊗_{R} k equals Y and the base change of f_{R} equals f. The intersection (Y_{R})^{f_{R}} of the graph of f_{R} and the diagonal of Y_{R} ×_{R} Y_{R} is the fixed subscheme of f_{R} (actually its image under the diagonal morphism). Since (Y_{R})^{f_{R}} is a proper scheme over Spec R, the image in Spec R is a closed subscheme of Spec R. To prove this closed subscheme equals all of Spec R, and thus contains the image of Spec k, it suffices to prove it contains a Zariski dense set of closed points.

Choose an f-invariant very ample sheaf, choose a basis for the space of global sections, and let A be the N × N matrix with entries in R giving the action of f on global sections with respect to this basis. The set of maximal ideal in Spec R with residue field of characteristic p > N are Zariski dense in Spec R. Every invertible matrix over a characteristic p field with order divisible by p^{2} has a Jordan block with eigenvalue 1 and size divisible by p. Thus, since p > N, the finite order of f_{R} modulo the prime is not divisible by p^{2}. Therefore, by the previous case, the reduction of f_{R} modulo the prime has nonempty fixed locus. Therefore the original automorphism f has nonempty fixed locus. □

This fixed point theorem implies that separably rationally connected varieties are simply connected. When the field k is ℂ, this was first proved by Campana using analytic methods, cf. the excellent reference by Debarre, [Deb01, Corollary 4.18].

Corollary 7.4 (Campana, Kollár). [Cam91], [Deb03, 3.6] Let X be a smooth, projective, separably rationally connected variety over an algebraically closed field k. The algebraic fundamental group of X is trivial. If k = ℂ, then the topological fundamental group of X is also trivial.

Kollár has generalized this considerably to prove a result for open subschemes of rationally connected varieties, cf. [Kol03].

Proof. The full proof is included in the beautiful survey by Debarre, [Deb03, 3.6]. Here is a brief sketch. First of all, for every quasi-projective, (not necessarily separably) rationally chain connected variety, Campana proved that the algebraic fundamental group is finite and also the topological fundamental group is finite when k equals ℂ (so that the topological fundamental group is defined). Thus the universal cover X̃ → X is finite. Since X is smooth, projective and separably rationally connected, also X̃ is smooth, projective and separably rationally connected. If the fundamental group of X is nonzero, then it contains a cyclic subgroup ℤ/nℤ such that p^{2} does not divide n. Of course the action of this group on X̃ is fixedpoint-free. But Corollary 7.3 implies there exists a fixed point. Thus X is simply connected. □

Theorem 1.9 also plays an important role in the proof of a “converse” to that same theorem.

Theorem 7.5. [GHMS05] Let π : X → B be a surjective morphism of normal, projective, irreducible varieties over an algebraically closed field k of characteristic zero. Assume that for some sufficiently large, algebraically closed field extension K/k, for every k-morphism C → B from a smooth, projective, K-curve to B, the pullback π_{C} : C ×_{B} X → C has a section. Then there exists a closed subvariety Y ⊂ X such that the geometric generic fiber of π|_{Y} : Y → B is nonempty, irreducible and rationally connected.

One corollary of this theorem, in fact the motivation for proving it, was to answer a question first asked by Serre and left unresolved by Theorem 1.9: could it be that a smooth, projective variety X over the function field of a curve has a rational point if it is O-acyclic, i.e., if h^{i}(X, O_{X}) equals 0 for all i > 0? One reason to ask this is that the corresponding question has a positive answer if “function field” is replaced by “finite field” thanks to N. Katz’s positive characteristic analogue of the Atiyah-Bott fixed point theorem, [DK73, Exposé XXII, Corollaire 3.2], recently generalized by Esnault, [Esn03]. Nonetheless, the answer is negative over function fields.

Corollary 7.6. [GHMS05] There exists a surjective morphism π : X → B of smooth, projective varieties over ℂ such that B is a curve and the geometric generic fiber of π is an Enriques surface, but π has no section. Thus, to guarantee a fibration over a curve has a section, it is not sufficient to assume the geometric generic fiber is O-acyclic.

In fact G. Lafon found an explicit morphism π as in Corollary 7.6 where B is ℙ_{ℂ}^{1}, or in fact ℙ_{k}^{1} for any field k with char(k) ≠ 2, and there does not even exist a power series section near 0 ∈ ℙ_{k}^{1}, cf. [Laf04].

CHAPTER 4 The Period-Index theorem of de Jong

1. Statement of the theorem

This chapter and the next are extracted from an earlier set of lecture notes, [Sta08]. The results in this chapter are joint work with A. J. de Jong. In particular, a full proof of Theorem 4.2 is presented in [Sta08]. For this reason we will not reproduce the proof here, but we will reproduce enough of the proof to prove de Jong’s Period-Index theorem.

Recall Problem 4.15: for a given field K, what is the precise relationship between the index of division algebras and the period, or exponent, of division algebras? Of course if K is algebraically closed, or more generally if K has cohomological dimension ≤ 1, Theorem 3.3(i) implies that the only division algebra over K is K itself. Thus both the period and the index equal 1. And in this sense the power r(K) of K equals 0. In particular, by Chevalley’s theorem and Tsen’s theorem discussed in Chapter 2, this holds if K is a finite field or if K is the fraction field of a curve over an algebraically closed field. The following question was explicitly asked by Colliot-Thélène, cf. [CT01].

Question 1.1. Let K be the function field of a variety of dimension d ≥ 2 over ℂ. For every division algebra D over K, does index(D) divide period(D)^{d−1}? In other words, is r(K) equal to d − 1?

There are examples showing r(K) is ≥ d−1, so this question is sharp. One can also ask this question with ℂ replaced by any algebraically closed field k, perhaps with the restriction that char(k) does not divide period(D). Even more generally, one could ask this question with K replaced by any C_{d} field, e.g., a function field of a variety of dimension d − 1 over a finite field.

The first major advance in this direction (from which subsequent advances have flowed) is the following theorem of A. J. de Jong.

Theorem 1.2 (de Jong’s Period-Index theorem). [dJ04] Let k be an algebraically closed field. Let K/k be the function field of a surface over k. Then the power of K is 1, i.e., for every central simple algebra A over K, the period of A equals the index of A.

Initially de Jong proved this under the additional hypothesis that char(k) does not divide the period of A. Using the technique of discriminant avoidance, described below, de Jong and I removed this hypothesis, cf. [dJS05, Theorem 1.0.2]. The original proof of de Jong used an analysis of the moduli space of Azumaya algebras, i.e., sheaves of central simple algebras, on a smooth, projective surface. The second proof of de Jong and myself will be presented below. Lieblich used moduli spaces of α-twisted sheaves, as introduced by Căldăraru and Yoshioka, to give a third proof. Moreover, Lieblich’s proof proves more.

Theorem 1.3. [Lie08] For K the function field of a surface over a finite field k, for every division algebra D over K, if char(k) does not divide period(D), then index(D) divides period(D)^{3}. If, moreover, D is the generic fiber of an Azumaya algebra on a smooth, proper model of K, then index(D) equals period(D).

Before explaining the second proof due to de Jong and myself, there are 3 more reformulations of Theorem 1.2. The first reformulation has to do with the map Δ_{n} from Theorem 3.15.

Lemma 1.4. For a field K, the power r(K) equals 1, i.e., the period equals the index for every division algebra over K, if and only if for every integer n ≥ 1 the map

Δ_{n} : H^{1}(Gal(K^{s}/K), PGL_{n,K}(K^{s})) → Br′(K)[n]

is surjective, and thus bijective.

Proof. Assume first that r(K) equals 1. By Theorem 4.7 and Theorem 4.12, every element of Br′(K) is of the form Δ[D] for some division algebra D. Since r(K) equals 1, the order of Δ[D] equals the index of D. Thus, if the order r of Δ[D] divides n, say n = rl, then A = D ⊗_{K} Mat_{l×l,K} gives an element of H^{1}(Gal(K^{s}/K), PGL_{n,K}(K^{s})) mapping to Δ[D] in Br′(K)[n]. Thus Δ_{n} is surjective.

Next assume that Δ_{n} is surjective for every n. Let D be a division algebra over K, and denote by n the order of Δ(D). Since Δ_{n} is surjective, there exists an Azumaya algebra A with dim_{K}(A) = n^{2} such that Δ_{n}(A) equals Δ(D). By Theorem 4.7, A is Morita equivalent to D. By Theorem 4.12, A = D ⊗_{K} Mat_{l×l,K} for some integer l ≥ 1. Comparing dimensions of these K-vector spaces, the period n is at least as large as the index of D. Since the period divides the index, also the index is at least as large as the period, and thus both are equal. □

The second reformulation requires the auxiliary notion of generalized Brauer-Severi varieties. Let A be a central simple algebra over K with dim_{K}(A) = n^{2} and let r ≥ 1 be an integer. The functor

K−algebras → Sets, R ↦ {I ⊂ A⊗_{K}R| I a free R-module of rank rn and a right ideal }

is representable.

Definition 1.5. [Bla91], [SVdB92] The generalized Brauer-Severi variety, Grass_{r,A}, is the scheme representing the functor above. Stated differently, if E denotes the PGL_{n,K}-torsor associated to A, then Grass_{r,A} equals the twist_{E}Grass_{K}(r, K^{⊕n}) of the usual Grassmannian Grass_{K}(R, K^{⊕n}) of locally free, rank r, locally direct summands of K^{⊕n}.

Lemma 1.6. For a field K, the power r(K) equals 1 if and only if for every central simple algebra A, for r = period(A), Grass_{r,A} has a K-point.

Proof. Assume first that r(K) equals 1. Let A be a central simple algebra with dim_{K}(A) = n^{2}. By Theorem 4.12, A is isomorphic to D ⊗_{K} Mat_{l×l,K} for some division algebra D over K and some integer l ≥ 1, i.e.,

A = End_{D−mod}(D^{⊕l}).

Since r(K) equals 1, period(A) = period(D) equals index(D). Denote by D^{⊕1} ⊂ D^{⊕l} the left D-submodule of D^{⊕l} generated by the first basis element. Then

I = Hom_{D−mod}(D^{⊕l}, D^{⊕1})

is a right ideal in A with dim_{K}(I) = rn. Thus I gives a K-point of Grass_{r,A}.

Next assume that Grass_{r,A} always has a K-point. Let D be a division algebra of order r. Since Grass_{r,D} has a K-point, there is a right ideal I in D with

dim_{K}(I) = period(D) · index(D).

Since the only nonzero right ideal in D is D itself, I equals D. Since dim_{K}(D) = index(D)^{2}, dividing gives

period(D) = index(D).

□

Among all the generalized Brauer-Severi varieties Grass_{r,A}, the one with r = period(A) has a special property.

Lemma 1.7. Let K be a field and let A be a central simple algebra over K with dim_{K}(A) = n^{2}. Let r be an integer, 1 ≤ r ≤ n. There exists an invertible sheaf L on Grass_{r,A} such that Pic(Grass_{r,A} ⊗_{K} K^{s}) = ℤ · (L ⊗_{K} K^{s}), i.e., the base change of L to Grass_{r,A} ⊗_{K} K^{s} = Grass_{K^{s}}(r, (K^{s})^{⊕n}) generates the Picard group, if and only if period(A) divides r.

Proof. Consider the commutative diagram of central extensions of smooth group schemes (similar to the one from the proof of Theorem 3.15).

1 ⟶ 𝔾_{m,K} ⟶  GL_{n,K}  ⟶ PGL_{n,K} ⟶ 1 ^{(−)^{r}} ↓  ↓^{q}  ↓^{=} 1 ⟶ 𝔾_{m,K} ⟶_{s} GL_{n,K}/μ_{r,K} · Id_{n×n} ⟶ PGL_{n,K} ⟶ 1.

Using Proposition 2.16 and Remark 2.17, Δ_{n}(A) has order dividing r if and only if the associated PGL_{n,K}-torsor of A lifts to a torsor for GL_{n,K}/μ_{r,K}. On the other hand, the image of the natural homomorphism

⋀^{r} : GL_{n,K} → GL_{\binom{n}{r},K}

is precisely the quotient GL_{n,K}/μ_{r,K}. Thus, from the commutative diagram

1 ⟶ 𝔾_{m,K} ⟶_{s} GL_{n,K}/μ_{r,K} ⟶ PGL_{n,K} ⟶ 1
=↓  ↓^{⋀^{r}}  ↓^{⋀^{r}}
1 ⟶ 𝔾_{m,K} ⟶ GL_{\binom{n}{r},K} ⟶ PGL_{\binom{n}{r},K} ⟶ 1

a PGL_{n,K}-torsor lifts to a torsor for GL_{n,K}/μ_{r,K} if and only if the associated PGL_{\binom{n}{r},K}-torsor lifts to a torsor for GL_{\binom{n}{r},K}, i.e., if it lifts from the group of

projective equivalences on ℙ(K^{⊕\binom{n}{r}}) the group of equivalences of (ℙ(K^{⊕\binom{n}{r}}), O(1)). Since the restriction of O(1) to Grass_{K}(r, K^{⊕n}) is the generator of the Picard group, it follows that Δ_{n}(A) has order dividing r if and only if there exists an invertible sheaf L on Grass_{r,A} whose base change to K^{s} is a generator of the Picard group. □

This is relevant to Lemma 1.6 because of the elementary obstruction to the existence of K-points.

Definition 1.8. [CTS87] Let K be a field and let X be a geometrically integral, quasi-projective, smooth K-scheme. The elementary obstruction to the existence of a K-point of X is the short exact sequence of Gal(K^{s}/K)-modules,

1 ⟶ (K^{s})^{∗} ⟶ K^{s}(X)^{∗} ⟶ K^{s}(X)^{∗}/(K^{s})^{∗} ⟶ 1

where K^{s}(X) is the fraction field of X ⊗_{K} K^{s}. Stated differently, the elementary obstruction is zero if and only if there exists a Galois-invariant splitting of (K^{s})^{∗} → K^{s}(X)^{∗}.

Remark 1.9. If X has a K-point, or even a 0-cycle of degree 1 over K, then the elementary obstruction is zero, cf. [CTS87, Prop. 2.2.2], [Sko01, Theorem 2.3.4]. And if X is projective with Pic(X ⊗_{K} K^{s}) isomorphic to ℤ, then the elementary obstruction vanishes if and only if there exists an invertible sheaf L on X such that L ⊗_{K} K^{s} is a generator for the Picard group, cf. [BCTS08, Lemma 2.2(v)].

This leads to the final reinterpretation of Theorem 1.2

Lemma 1.10. For a field K, the power r(K) equals 1 if and only if for every central simple algebra A, denoting by n the integer such that dim_{K}(A) = n^{2}, for every integer r with 1 ≤ r ≤ n, vanishing of the elementary obstruction for Grass_{r,A} is a sufficient condition for existence of a K-point.

This last reformulation suggests a problem, the refined version of Problem 1.1 for function fields of surfaces.

Problem 1.11. Let K be the function field of a surface over an algebraically closed field k. Let X be a smooth, projective K-scheme. Find sufficient conditions on X ⊗_{K} K^{s} so that X has a K-point when the elementary obstruction vanishes, e.g., when there exists an invertible sheaf L on X such that L ⊗_{K} K^{s} is a generator for Pic(X ⊗_{K} K^{s}).

2. Abel maps over curves and sections over surfaces

Let κ be a characteristic 0 field, not necessarily algebraically closed. Let C be a smooth, projective, geometrically integral curve over κ. Let π : X → B be a projective, flat morphism whose geometric generic fiber is irreducible and smooth. And let L be a π-ample invertible sheaf on X.

Definition 2.1. For a κ-scheme S, a family of sections of π parameterized by S is a morphism of C-schemes

τ : S ×_{κ} C → X.

For an integer e, the family of sections has degree e if the invertible sheaf τ^{∗}L on S ×_{κ} C has relative degree e over S, i.e., for every geometric point s of S, the basechange of the invertible sheaf to C_{s} has degree e. A pair (S, τ) as above is universal if for every κ-scheme S′ and for every family of degree e sections of π parameterized by S,

τ′ : S′ ×_{κ} C → X,

there exists a unique κ-morphism f : S′ → S such that τ′ equals τ ∘ (f, Id_{C}).

Theorem 2.2 (Grothendieck). [Gro62, Part IV.4.c, p. 221-19] For every integer e there exists a pair (Sections^{e}(X/C/κ), σ) of a κ-scheme and a family of degree e sections of π parameterized by Sections^{e}(X/C/κ),

σ : Sections^{e}(X/C/κ) ×_{κ} C → X,

such that σ^{∗}L which is universal. Moreover Sections^{e}(X/C/κ) is a quasi-projective κ-scheme.

Invertible sheaves on C of degree e are parameterized by the Picard scheme Pic_{C/κ}^{e}. Thus, associated to the invertible sheaf σ^{∗}L there is a morphism of κschemes

α : Sections^{e}(X/C/κ) → Pic_{C/κ}^{e}.

This morphism is the Abel map associated to L.

The quasi-projective κ-scheme Sections^{e}(X/C/κ) is rarely projective. There is a projective scheme Σ^{e}(X/C/κ) containing Sections^{e}(X/C/κ) as an open subscheme: the coarse moduli space of the stack of stable sections, i.e., stable maps to X whose stabilization under π is an isomorphism to C. Stable maps are an important tool in this area, but a discussion of the basics of stable maps would carry us rather far from the main story. Thus, suffice it to say that Σ^{e}(X/C/κ) is a projective scheme, and there exists a κ-morphism

α : Σ^{e}(X/C/κ) → Pic_{C/κ}^{e}.

whose restriction to Sections^{e}(X/C/κ) is the Abel map from the last paragraph. Although the following problem may at first appear to have little to do with Problem 1.11, in fact the two are intimately connected.

Problem 2.3. Let κ, π : X → C and L be as above. Assume that the base change of L to the geometric generic fiber of π generates the Picard group. Find conditions conditions under which there exists a sequence of closed subschemes (Z_{e} ⊂ Σ^{e}(X/C/κ)_{e≫0} such that Z_{e} intersects Sections^{e}(X/C/κ) and such that the restriction of the Abel map

α|_{Z_{e}} : Z_{e} → Pic_{C/κ}^{e}

is surjective with rationally connected geometric generic fiber.

In fact there do exist such conditions which capture, in some sense, the notion of “rational simple connectedness”.

3. Rational simple connectedness hypotheses

Hypothesis 1. The κ-scheme X is smooth and projective.

Hypothesis 2. The invertible sheaf L is π-ample and π-relatively globally generated.

All of the hypotheses are insensitive to replacing L by L ⊗_{O_{X}} π^{∗}M for an invertible sheaf M on B. Thus, in Hypothesis 2, we may even assume L is ample and globally generated.

The remaining hypotheses are stated in terms of Y.

Hypothesis 3. The basechange L_{Y} of L to Y is very ample.

From now on we consider Y to be embedded in a projective space ℙ_{K}^{N} by the complete linear system of L_{Y}. In particular, \overline{M}_{0,1}(Y/K, 1) is the projective Kscheme parameterizing pointed lines (p, L) in Y, i.e., L is a line contained in Y and p is a point of L. There is an “evaluation” K-morphism

ev : \overline{M}_{0,1}(Y/K, 1) → Y, (L, p) ↦ p.

Hypothesis 4. The morphism ev above is surjective, the geometric generic fiber of ev is irreducible and rationally connected, and the fiber of ev over every codimension 1 point η of Y is “Morse”, i.e., \overline{M}_{0,1}(Y/K, 1) is smooth at every point of ev^{−1}(η) and the only singularities on the geometric fiber over η are at worst finitely many ordinary double point singularities.

Of course there is a dense open subset of Y, Y_{free}, such that every line in Y intersecting Y_{free} is free. Thus

ev : ev^{−1}(Y_{free}) → Y_{free}

is smooth and projective. In particular, the fiber of ev over the geometric generic point of Y is automatically smooth.

If one thinks of rational curves in Y as analogous to continuous paths in a CW complex, this hypothesis is analogous to asking that the space of continuous paths with one boundary point fixed is itself path-connected. Since continuous paths with one fixed boundary point can be retracted to a constant path to the basepoint, the condition always holds. The Morse condition is technical, but often easy to verify. Notice also that this hypothesis implies that L_{Y} is not itself a tensor power K^{⊕d} for an integer d > 1.

The next hypothesis uses the space of chains of free lines. For an integer n ≥ 1, a chain of n free lines is an n-tuple

(C, p_{1}, q_{n}) = ((L_{1}, p_{1}, q_{1}), (L_{2}, p_{2}, q_{2}), …, (L_{n}, p_{n}, q_{n}))

where each L_{i} is a free line in Y, p_{1} and q_{i} are K-points of L_{i}, and q_{i} = p_{i+1} for i = 1, …, n − 1. The points p_{1} and q_{n} are considered “boundary points” of the chain analogous to the boundary points 0 and 1 of the closed unit interval [0, 1].

There is a quasi-projective K-scheme FreeChain_{2}(Y/K, n) parameterizing chains of n free lines. And there is an “evaluation” K-morphism

ev : FreeChain_{2}(Y/K, n) → Y ×_{K} Y, (C, p_{1}, q_{n}) ↦ (p_{1}, q_{n}).

Hypothesis 5. For some positive integer n = n_{0}, the morphism ev above is dominant and the geometric generic fiber is “birationally rationally connected”, i.e., it is isomorphic as a scheme over \overline{K(Y ×_{K} Y)} to an open subscheme of a projective, rationally connected scheme.

Under Hypotheses 3 and 4, ev is dominant if and only if Y is irreducible and L_{Y} is a generator of Pic(Y). If the Picard number equals 1, the argument proving ev is dominant appears, for instance, in [Kol96, Corollary IV.4.14]. And the opposite direction follows from [Kol96, Proposition IV.3.13.3].

Together with Hypotheses 1 and 2, this implies a condition on the singular fibers of π: The specializations of the geometric generic fiber occurring as singular fibers of π are sufficiently mild that the ample generator of the Picard group of the geometric generic fiber extends to an ample invertible sheaf on the special fiber. In particular this implies all singular geometric fibers are irreducible and more.

The hypothesis that the fibers of ev are rationally connected is very strong. The fiber of ev is analogous to the space of continuous paths in a path connected CW complex connecting two points. The CW complex is simply connected if and only if this path space is simply connected. In this way we consider Hypothesis 5 to be a version of “rational simple connectedness”.

The final hypothesis may seem technical, but it is essential. There are examples where Hypotheses 1 through 5 hold and yet the fibers of the Abel maps are not rationally connected.

Hypothesis 6. There exists a very twisting scroll in Y, i.e., a pair (R, D) consisting of a closed subscheme R of ℙ_{K}^{1} ×_{K} Y and a Cartier divisor class D on R such that

(i) R is a scroll, i.e., pr_{ℙ^{1}} : R → ℙ^{1} is a smooth morphism whose geometric fibers are lines in Y,

(ii) O_{R}(D) is globally generated and D has relative degree 1 over ℙ^{1}, (iii) the normal sheaf N_{R/ℙ^{1}×Y} is globally generated, and (iv) h^{1}(R, N_{R/ℙ^{1}×Y} (−D) ⊗_{O_{ℙ^{1}}} O_{ℙ^{1}}(−2)) equals 0. In fact this complicated condition has a simple geometric meaning. By Bertini’s

theorem, a general member of the linear system |D| is a smooth curve in R. By (ii), this curve is of the form τ(ℙ^{1}) for a section τ : ℙ^{1} → R of pr_{ℙ^{1}}. The scroll R together with the section τ defines a family of pointed lines in Y parameterized by ℙ_{K}^{1}. Equivalently, it defines a K-morphism

ζ : ℙ_{K}^{1} → \overline{M}_{0,1}(Y, 1).

The conditions (ii)–(iv) say that ζ(ℙ^{1}) is in the smooth locus of both morphisms

ev : \overline{M}_{0,1}(Y, 1) → Y

and

ρ : \overline{M}_{0,1}(Y, 1) → \overline{M}_{0,0}(Y, 1),

that the pullback ζ^{∗}T_{ρ} := ζ^{∗}(Ω_{ρ})^{∨} is globally generated, and that the pullback ζ^{∗}T_{ev} := ζ^{∗}(Ω_{ev}^{∨} is ample. In fact if there exists a morphism ζ with these properties, then there also exists a morphism such that both ζ^{∗}T_{ρ} and ζ^{∗}T_{ev} are ample. Then ζ is very free relative to both ev and ρ.

4. Rational connectedness of the Abel map

These are all the hypotheses. However, before stating the theorem there is one more definition.

Definition 4.1. Let d ≥ 0 be an integer. For an algebraically closed extension field k of κ, a d-free section of π defined over k is a section σ : C ⊗_{κ} k → X defined over k such that for one (and hence every sufficiently general) effective Cartier divisor D of C ⊗_{κ} k of degree d,

h^{1}(C ⊗_{κ} k, σ^{∗}N_{σ(C⊗_{κ}k)/X⊗_{κ}k}(−D)) equals 0.

In particular, when d ≥ max 2g, 1 the first cohomology vanishes if and only if the map on global sections

H^{0}(C ⊗_{κ} k, σ^{∗}N_{σ(C⊗_{κ}k)/X⊗_{κ}k}) → H^{0}(C ⊗_{κ} k, σ^{∗}N_{σ(C⊗_{κ}k)/X⊗_{κ}k} ⊗_{O_{C}} O_{D})

is surjective. We call a section (g)-free if it is d-free for some d ≥ max 2g, 1, i.e., for g ≥ 1 it is 2g-free and for g = 0 it is 1-free.

Theorem 4.2 (de Jong, Starr). If (X, L, π) satisfies Hypotheses 1 through 6, then there exists a sequence (Z_{e})_{e≫0} of irreducible components Z_{e} of Σ^{e}(X/C/κ) such that

(i) for each e ≫ 0, a general point of Z_{e} parameterizes a free section of π, (ii) for each e ≫ 0, the Abel map

α|_{Z_{e}} : Z_{e} → Pic_{C/κ}^{e}

is surjective with rationally connected geometric generic fiber, and (iii) for every algebraically closed field extension k of κ, for every (g)-free section σ_{0} : C ⊗_{κ} k → X of π defined over k, and for all δ ≫ 0, every stable section obtained by attaching δ free lines in fibers of π to σ_{0}(C) gives a k-point of Σ^{e_{0}+δ}(X/C/κ) lying in Z_{e_{0}+δ}.

Moreover every geometric fiber X_{t} of π is integral. And every point of the smooth locus (X_{t})_{smooth} is contained in a very free rational curve in (X_{t})_{smooth}.

Note that (iii) implies that the sequence (Z_{e})_{e≫0} is canonical. In particular, it is Galois invariant. Thus, in order to verify the theorem over every characteristic 0 field κ, it suffices to verify the theorem over algebraically closed, characteristic 0 fields. Moreover, as the statements are geometric, it suffices to pass to “sufficiently large” algebraically closed fields, e.g., uncountable fields. This will be technically convenient later (although, of course, the interested reader can easily formulate all of our arguments over any characteristic 0 field). From this point on, we will assume that κ is an uncountable, algebraically closed field of characteristic 0.

The proof of this theorem goes beyond what we can present here. However we will present a proof in the special case necessary for de Jong’s Period-Index theorem.

5. Rational simply connected fibrations over a surface

Theorem 4.2 is important because of its application to the existence of rational sections of fibrations over surfaces.

Corollary 5.1 (de Jong, Starr). Let k be an algebraically closed field of characteristic 0. Let S be a smooth, integral, projective surface over k. Let f : X → S be a proper morphism. Assume there exists a Zariski open subset U of S and an invertible sheaf L on f^{−1}(U) such that

(i) S − U is a finite collection of k-points of S, (ii) f^{−1}(U) is smooth, (iii) the restriction

f|_{f^{−1}(U)} : f^{−1}(U) → U

is flat and surjective (iv) L is f-ample and f-relatively globally generated, and

(v) the geometric generic fiber Y of f together with the base change L_{Y} of L

satisfy Hypotheses 3–6 of Section 3.

Then there exists a rational section of f.

Proof. There exists a Lefschetz pencil of ample divisors on S whose basepoints are all contained in U. So after replacing S by the blowing up of the base locus, replacing f and L by the pullbacks over the blowing up, and replacing U by its inverse image in the blowing up, we may assume in addition that there exists a surjective, projective morphism

r : S → ℙ_{k}^{1}.

Theorem 2.2 generalizes to give a pair (Sections^{e}(X/S/ℙ_{k}^{1}), σ) of a ℙ_{k}^{1}-scheme Sections^{e}(X/S/ℙ_{k}^{1}) and an S-morphism

σ : Sections^{e}(X/S/ℙ_{k}^{1}) ×_{ℙ^{1}} S → X

which is universal among such pairs. And there is a generalization of the definition of Σ^{e}(X/C/κ) giving a ℙ^{1}-scheme Σ^{e}(X/S/ℙ_{k}^{1}) and a family of stable sections

(ρ : C → Σ_{e}(X/S/ℙ_{k}^{1}), h : C → X)

of π having L-degree e. And again using the methods of [KM76], there is an Abel map of ℙ_{k}^{1}-schemes

α : Σ_{e}(X/S/ℙ_{k}^{1}) → Pic_{S/ℙ_{k}^{1}}^{e},

at least when we restrict over the dense open subset (ℙ_{k}^{1})_{ω,good} – the maximal open subset over which ω is smooth (over the finitely many k-points of ℙ_{k}^{1} − (ℙ_{k}^{1})_{ω,good} there are some technical difficulties in representing the relative Picard functor by a projective scheme).

Denote the function field k(ℙ_{k}^{1}) by κ (of course it is not algebraically closed). So Spec κ → ℙ_{k}^{1} is the generic point of ℙ_{k}^{1}. The fibers of these ℙ_{k}^{1} schemes and morphisms of ℙ_{k}^{1}-schemes over Spec κ are

Σ^{e}(X/S/ℙ_{k}^{1}) ×_{ℙ_{k}^{1}} Spec κ = Σ^{e}(X_{κ}/C/κ),
Pic^{e}(X/S/ℙ_{k}^{1}) ×_{ℙ_{k}^{1}} Spec κ = Pic_{C/κ}^{e}

where X_{κ} denotes X ×_{ℙ_{k}^{1}} Spec κ and C denotes S ×_{ℙ_{k}^{1}} Spec κ. Moreover the basechange of the ℙ_{k}^{1}-morphism α above is the Abel map

α : Σ^{e}(X_{κ}/C/κ) → Pic_{C/κ}^{e}

defined in Section 2.

The hypotheses on X, L, U and Y imply Hypotheses 1–6 for X_{κ}, L_{κ} and Y (which hasn’t actually changed). Thus, by Theorem 4.2, there exists a sequence (Z_{e})_{e≥e_{0}} of irreducible components Z_{e} of Σ^{e}(X_{κ}/C/κ) satisfying Conditions (i)–(iii) of Theorem 4.2. For every e ≥ e_{0}, the closure of Z_{e} in Σ^{e}(X/S/ℙ_{k}^{1}) is an irreducible component Z_{ℙ_{k}^{1},e} of Σ^{e}(X/S/ℙ_{k}^{1}). Because of Conditions (i) for (Z_{e})_{e≥e_{0}}, a general point of Z_{ℙ_{k}^{1},e} intersects Sections^{e}(X/S/ℙ_{k}^{1}) is a dense open subset of Z_{ℙ_{k}^{1},e}. And because of Condition (ii), for e ≫ 0 the Abel map

α : Z_{ℙ_{k}^{1},e} → Pic_{S/ℙ_{k}^{1}}^{e}

is surjective (at least over (ℙ_{k}^{1})_{ω,good}), and the geometric generic fiber is rationally connected.

For every integer e there exist sections

σ : (ℙ_{k}^{1})_{ω,good} → Pic_{S/ℙ_{k}^{1}}^{e}.

Indeed, the exceptional divisor over the blowing up of every basepoint of the Lefschetz pencil of divisors on S is a Cartier divisor on S which defines a section to Pic_{S/ℙ_{k}^{1}}^{1}. So e times this section gives a section σ.

For e ≫ 0, the geometric generic fiber of

α : Z_{ℙ_{k}^{1},e} → Pic_{S/ℙ_{k}^{1}}^{e}

is rationally connected. It may happen, nonetheless, that the fiber of

α : Z_{ℙ_{k}^{1},e} → Pic_{S/ℙ_{k}^{1}}^{e}

over the geometric generic point of σ(ℙ_{k}^{1}) is not rationally connected. There is a standard way to deal with this issue. We can always realize σ(ℙ_{k}^{1}) as an irreducible component of multiplicity 1 in a complete intersection curve C_{0} in Pic_{S/ℙ_{k}^{1}}^{e}, i.e., C_{0} is a curve which is a complete intersection of g = g(C) very ample divisors in Pic_{S/ℙ_{k}^{1}}^{e}. And we can deform these divisors to obtain a family of complete intersection curves (C_{t}) in Pic_{S/ℙ_{k}^{1}}^{e} parameterized by a DVR R = k \begin{matrix}[ \\ [ & t & ] \\ ] \end{matrix}. By deforming the divisors appropriately, we can arrange that the fiber of C_{t} over the geometric generic point \overline{k \begin{matrix}( \\ ( & t & ) \\ ) \end{matrix}} is smooth, and the fiber of α over the geometric generic point of C_{t} is rationally connected.

Thus after performing a base change from k to L = \overline{k \begin{matrix}( \\ ( & t & ) \\ ) \end{matrix}}, the restriction of α over the geometric generic fiber

C_{L} := C ⊗_{k[[t]]} \overline{k \begin{matrix}( \\ ( & t & ) \\ ) \end{matrix}}

is a rationally connected fibration over the smooth, projective curve C_{L} over the algebraically closed field L. Thus, by Theorem 1.9, there exists a section. But by the usual limit theorems, every section over L can, in fact, be defined over the fraction field k(R′) = k \begin{aligned}( \\ ( & t^{1/N} & ) \\ & ) \end{aligned} of some finite extension R′ = k \begin{aligned}[ \\ [ & t^{1/N} & ] \\ & ] \end{aligned} of R.

Since σ(ℙ_{k}^{1}) is an irreducible component of multiplicity 1 in C_{0}, σ(ℙ_{k}^{1}) is still an irreducible component of multiplicity one in the base change C_{t} ⊗_{R} R′. Thus C_{t} ⊗_{R} R′ is normal at the generic point of σ(ℙ_{k}^{1}), which is a codimension 1 point of this 2-dimensional scheme. Finally, by the valuative criterion of properness applied to the proper ℙ^{1}-scheme Z_{ℙ_{t}^{1},e}, the section defined over C_{t} ⊗_{R} k(R′) extends to the generic point of σ(ℙ_{k}^{1}). Thus there exists a rational section

τ : σ(ℙ_{k}^{1}) −−→ Z_{ℙ_{k}^{1},e}

of α. Again by the valuative criterion of properness, this extends to all of σ(ℙ_{k}^{1}).

It may happen that τ(Spec κ) parameterizes a stable section rather than a section. But every stable section has a unique handle, i.e., the unique component which is the image of a section. Because the handle is canonical, it satisfies the cocycle condition for Galois descent, and thus is defined over the same field of definition as the stable section – in this case κ. Thus finally, quite possibly for some integer e′ < e, there exists a κ-point of Sections^{e′}(X/S/ℙ_{k}^{1}). By the universal property of Sections^{e′}(X/S/ℙ_{k}^{1}), this is equivalent to a morphism

Spec κ ×_{ℙ_{k}^{1}} S → X.

The closure of the image of this morphism is an integral closed subscheme Z of X such that the projection Z → S is a proper morphism which is an isomorphism over C. Thus it is birational, i.e., Z is the closure of the image of a rational section of X → S. Therefore f : X → S has a rational section. □

6. Discriminant avoidance

Even though the theorems above require that char(κ) equals 0, there is a technique for “lifting” from positive characteristic. It is recalled very briefly below, but the proof will not be discussed.

Let G be a reductive group scheme over some base scheme T. Let X be a smooth, projective T -scheme on which G acts. For every T -scheme S and every G-torsor T over S, there is an associated S-scheme

X_{T} := X ×_{T} T /G,

the quotient by the free action of G. Let U be a dense open subscheme of T. Let c be a nonnegative integer. Consider the following two properties of the datum (T, G, X) and the integer c.

Property 6.1. For every algebraically closed field over U, Spec k → U, for every projective, integral k-scheme S of dimension c, and for every G-torsor T over S, the projection

X_{T} → S

admits a rational section.

Property 6.2. For every algebraically closed field over T, Spec k → U, for every quasi-projective, integral k-scheme S of dimension c, and for every G-torsor T over S, the projection

X_{T} → S

admits a rational section.

The basic technique of “discriminant avoidance” proves the following.

Proposition 6.3. [dJS05] If Property 1 holds, then Property 2 holds.

Corollary 6.4. To prove Theorem 1.2, it suffices to prove Theorem 4.2 in the special case that the morphism

X ⊗_{κ} \overline{κ} → C ⊗_{κ} \overline{κ}

is of the form Grass_{C⊗_{κ}\overline{κ}}(r, E) for a locally free sheaf E of rank n divisible by r on C ⊗_{κ} \overline{κ}.

Proof. By Lemma 1.6, to prove Theorem 1.2 it suffices to prove that for every pair of positive integers r, n with r dividing n, for every field K as in Theorem 1.2, for every central simple algebra A with dim_{K}(A) = n^{2} and period(A) dividing r, the generalized Brauer-Severi variety Grass_{r,A} has a K-point.

Define T = Spec ℤ_{p} and define U = Spec ℚ_{p}. Let G be the group scheme GL_{n,T} /μ_{r,T} appearing in the proof of Lemma 1.7. And let (X , L) be (Grass_{T} (r, O_{T}^{⊕n}), O(1)). By the proof of Lemma 1.7, for every algebraically closed field k, for every fraction field K of a surface over k, and for every central simple algebra A over K with dim_{K}(A) = n^{2} and period(A) dividing r, there exists a quasi-projective surface S over k with fraction field K, and there exists a G-torsor T over S such that Grass_{r,A} is the generic fiber of the S-scheme X_{T}. Thus by Proposition 6.3, to prove Grass_{r,A} always has a K-point, it suffices to prove that for every characteristic 0 algebraically closed field k, for every projective surface S over k, and for every G-torsor T over S, X_{T} has a rational section over S. Define L on X_{T} to be the twist of L on X.

By resolution of singularities on surfaces in characteristic 0 (which is reasonably easy), we may as well assume S is smooth. Then the proof of Corollary 5.1 reduces existence of a rational section to Theorem 4.2 for the restriction of X_{T} and L to the generic fiber C of a Lefschetz pencil of divisors on S. By Corollary 2.12 and by Proposition 3.2(i), the base change of T to C ⊗_{κ} \overline{κ} lifts to a torsor for GL_{n,\overline{κ}}. This is equivalent to a locally free sheaf E of rank n on C ⊗_{κ} \overline{κ}. Thus the base change of X_{T} to C ⊗_{κ} \overline{κ} equals Grass_{C⊗_{κ}\overline{κ}}(r, E). □

7. Proof of the main theorem for Grassmann bundles

We will not prove Theorem 4.2, but we will prove it for Grassmann bundles coming from Azumaya algebras over C. The following proof is an ad hoc argument using the well-known fact that the moduli space of stable sheaves of fixed rank and determinant on a curve is unirational.

Proposition 7.1. Let κ be a characteristic 0 field, let C be a smooth, projective, geometrically integral κ-scheme, let

π : X → C

be a flat, projective morphism, and let L be a π-ample invertible sheaf on X whose base change to the geometric generic fiber of π generates the Picard group. Assume further that the morphism

π ⊗ Id : X ⊗_{κ} \overline{κ} → C ⊗_{κ} \overline{κ}

is a Grassmann bundle Grass_{C⊗_{κ}\overline{κ}}(r, E) for a locally free sheaf E of rank n divisible by r on C ⊗_{κ} \overline{κ}. Then the conclusion of Theorem 4.2 holds. Therefore, by Corollary 6.4, de Jong’s Period-Index theorem, Theorem 1.2, also holds.

Proof. Thus Sections^{e}(X/C/κ) ⊗_{κ} \overline{κ} is the scheme parametrizing locally free, rank r quotients Q of E^{∨} of degree e. And the Abel map associates to Q the determinant det(Q) = ⋀^{r} Q (up to a constant translation which makes no difference in the following).

Denote by V_{e} the subscheme V_{e} of Sections^{e}(X/C/κ) ⊗_{κ} \overline{κ} parametrizing quotients with are stable sheaves. Since stability is an open condition in families, V_{e} is an open subscheme. Moreover, since Q is stable if and only if every twist of Q by an invertible sheaf is stable, V_{e} is independent of the choice of lift of the torsor (T ×_{S} C) ⊗_{κ} \overline{κ} to a GL_{n,L}-torsor over C ⊗_{κ} L, where T is as in the proof of Corollary 6.4. Therefore V_{e} is invariant under the action of Gal(\overline{κ}/κ), i.e., V_{e} equals U_{e} ⊗_{κ} \overline{κ} for some open subscheme U_{e} of Sections^{e}(X/C/κ). Finally, for e ≫ 0, there exist quotients Q which are stable so that U_{e} is nonempty. Define Z_{e} to be the closure of U_{e}.

By standard arguments it is not hard to show that for e ≫ 0 every stable sheaf of rank r and degree e on C ⊗_{κ} L appears as a quotient Q, and also that

h^{1}(C, Hom_{O_{C}⊗_{κ}L}(E, Q) equals 0.

Therefore the forgetful morphism

Φ : U_{e} → Bun_{r,e,stable}(C ⊗_{κ} L)

to the moduli space of stable rank r, degree e, locally free sheaves on C ⊗_{κ} L is surjective and the fiber over every point Q is an open subset of the constant rank affine space Hom_{O_{C}⊗_{κ}L}(E, Q), i.e., Φ is an open subscheme of an affine bundle over the moduli space of stable sheaves. And the Abel map factors through the “usual” Abel map

det : Bun_{r,e,stable}(C ⊗_{κ} L) → Pic_{C⊗_{κ}L/L}^{e}.

As is well-known, the morphism det above has rationally connected (and even unirational) fibers of dimension (r^{2} − 1)(g(C) − 1), i.e., the moduli space of stable sheaves of rank r and fixed determinant is unirational. Thus the geometric generic fiber of

α|_{U_{e}} : U_{e} → Pic_{C/κ}^{e}

is isomorphic to an open subscheme of an affine bundle over a rationally connected variety, which is itself an open subscheme of a rationally connected projective variety. But it is also isomorphic to a dense open subset of the projective variety which is the geometric generic fiber of

α|_{Z_{e}} : Z_{e} → Pic_{C/κ}^{e}.

Since rational connectedness is a birational invariant among projective varieties, the geometric generic fiber of α|_{Z_{e}} is rationally connected. □

CHAPTER 5 Rational simple connectedness and Serre’s “Conjecture II”

1. Generalized Grassmannians are rationally simply connected

This chapter is extracted from an earlier set of lecture notes, [Sta08]. The omitted proofs can be found there. The results of this chapter are joint work with A. J. de Jong and Xuhua He, heavily relying on a strategy proposed by Philippe Gille.

The proof of Proposition 7.1 used an ad hoc argument. But in fact one can directly verify the hypotheses of Theorem 4.2 for Grassmannians. The most difficult hypothesis to verify is the existence of a very twisting scroll. This is not too bad for Grassmannians. But for generalized Grassmannians, i.e., minimal (positive dimensional) homogeneous spaces, for other groups, e.g., groups of type E_{8}, it becomes unreasonable to verify this “by hand”. Thus it is extremely fortunate that Xuhua He has given a very simple, elegant proof that every generalized Grassmannian satisfies the hypotheses. Moreover, the proof uses only general results about root systems – it does not use the classification of simple algebraic groups. The conclusion is the following.

Corollary 1.1 (de Jong, He, Starr). Let k be an algebraically closed field of characteristic 0. Let X be a smooth, connected, projective k-scheme of positive dimension which is a homogeneous space for a linear algebraic group scheme G over k. Assume there exists an invertible sheaf L on X which is an ample generator for Pic(X). Then X and L satisfy Hypotheses 3–6 of Section 3.

Corollary 1.2 (de Jong, He, Starr). Let k be an algebraically closed field of characteristic 0. Let S be a smooth, integral, projective surface over k. Let f : X → S be a proper morphism. Assume there exists a Zariski open subset U of S and an invertible sheaf L on f^{−1}(U) such that

(i) S − U is a finite collection of k-points of S, (ii) f^{−1}(U) is smooth, (iii) the restriction

f|_{f^{−1}(U)} : f^{−1}(U) → U

is flat and surjective (iv) L is f-relatively globally generated, and

(v) the geometric generic fiber Y of f is a homogeneous space for a linear algebraic group over K = \overline{k(S)} and L_{Y} is an ample generator for Pic(Y).

Then there exists a rational section of f.

2. Statement of the theorem

Serre’s Conjecture II, Conjecture 5.5, has been proved for some important classes of fields, but in general it remains open. One class of fields satisfying the hypotheses of the conjecture is the class of function fields K of surfaces over an algebraically closed field k. Notice that if char(k) is positive, then K is not perfect but does satisfy the conditions of Remark 5.6.

For these fields, there has been tremendous progress. In particular, work of Merkurjev–Suslin, Ojanguren–Parimala, Colliot-Thélène–Gille–Parimala and Gille has reduced Serre’s Conjecture II for all function fields K of surfaces over an algebraically closed field k to the special case for such fields when G is an simple algebraic group of type E_{8}, cf. [CTGP04, Theorem 1.2(v)]. The main application of Theorem 4.2 and Corollary 5.1 settles the “split case” of Serre’s Conjecture II for function fields K.

Theorem 2.1 (de Jong, He, Starr). Let k be an algebraically closed field and let K/k be the function field of a surface. Let G be a connected, simply connected, semisimple algebraic group over k. Every G-torsor over K is trivial.

In particular, if G_{K} is a simple algebraic group over K of type E_{8}, then G_{K} is itself split and thus every G_{K}-torsor over K is trivial. So Serre’s Conjecture II holds over function fields for groups of type E_{8}.

Remark 2.2. As we explain further below, in addition to Corollary 1.2, the main input to Theorem 2.1 is a strategy due to Philippe Gille.

3. Reductions of structure group

There is a statement that implies both Theorem 2.1 as well as again proving Theorem 1.2 (it is essentially the same proof as above). The statement below is a generalization of a consequence of Theorem 4.2 proved by Philippe Gille. Gille’s strategy is absolutely fundamental in the proof of Theorem 2.1. Let k be an algebraically closed field. Let G be a (smooth) connected, simply connected, semisimple algebraic group over k, and let P be a (reduced) parabolic subgroup of G. The center Z_{G} of G is a finite group scheme which is contained in P. There is a maximal quotient P ↠ T_{P} which is an algebraic torus. Denote by Z_{G,P} the kernel of the induced homomorphism Z_{G} → T_{P}. The natural action of G on G/P lifts canonically to a linear action on every invertible sheaf over G/P. The finite subgroup scheme Z_{G,P} is the maximal subgroup acting trivially on G/P and on every invertible sheaf over G/P. Thus G/Z_{G,P} is the maximal quotient of G acting on G/P whose action lifts to a linear action on every invertible sheaf over G/P.

Theorem 3.1. Let K/k be the function field of a surface over k. For every torsor T for G/Z_{G,P} over K, the associated K-variety X = T /P has a K-point. Equivalently, the torsor T has a reduction of structure group to P/Z_{G,P}.

Proof. Let T denote Spec of the Witt ring of k, e.g., T = Spec k if k has characteristic 0 and T = Spec ℤ_{p} if k equals ℤ/pℤ. And denote by U the open subset of T which consists of the generic point only.

Associated to the root datum for G, we can construct a smooth, linear algebraic group scheme G_{T} over T. And associated to the parabolic P, we can construct a closed subgroup scheme P_{T} over T. The definition of Z_{G,P} extends to give a finite, flat group scheme Z_{G_{T} ,P_{T}} over T. The quotient group scheme G = G_{T} /Z_{G_{T} ,P_{T}} is a reductive group scheme since it is T -flat and the closed fiber is reductive. And the T -scheme X = G_{T} /P_{T} is smooth and quasi-projective. Since the closed fiber is proper over k, X is projective over T. Thus G and X satisfy the hypotheses in Section 6. The goal is to prove Property 6.2 for c = 2. Because of Proposition 6.3, it suffices to prove Property 6.1 for c = 2. In particular, to prove the theorem it suffices to assume that k has characteristic 0.

Now we use an induction argument proposed by Philippe Gille. The induction is on the corank, rank(G) − rank(P). The base case is when P is a maximal parabolic. Again applying Proposition 6.3, to prove the result for all G/Z_{G,P} - torsors over fraction fields K of surfaces over k, it suffices to prove the result for each torsor which is the generic fiber of a G/Z_{G,P} -torsor over a smooth, projective, connected surface S over k. In this case the S-scheme X_{S} = [T_{S} ×(G/P)]/(G/Z_{G,P}) satisfies the hypotheses of Corollary 1.2. Thus Corollary 1.2 implies the result in this case.

By way of induction, assume the corank is > 1 and the result is known for all smaller values of the corank. Since the corank is > 1, P is not a maximal parabolic. Let Q be a maximal parabolic containing P. Then Z_{G,P} is contained in Z_{G,Q}. For every G/Z_{G,P} -torsor over K, by the base case, the associated G/Z_{G,Q}-torsor has a reduction of structure group to Q/Z_{G,Q}. Thus the original G/Z_{G,P} -torsor has a reduction of structure group to Q/Z_{G,P} (observe (G/Z_{G,P})/(Q/Z_{G,P}) is the same as (G/Z_{G,Q})/(Q/Z_{G,Q}) since both are just G/Q).

Now Q has a filtration by normal subgroup schemes,

Q = Q_{0} ⊃ Q_{1} ⊃ Q_{2},

where Q_{2} is the unipotent radical of Q and where Q_{0}/Q_{1} is the maximal quotient of Q which is of multiplicative type, i.e., isomorphic to G_{m,k}. By Hilbert’s Theorem 90, every Q_{0}/Q_{1}-torsor over K is trivial, thus there is a reduction of structure group to Q_{1}/Z_{G,P} (by construction Z_{G,P} is contained in P ∩Q_{1}). And over a characteristic 0 field, every torsor for a unipotent group is trivial. Thus this Q_{1}/Z_{G,P} -torsor has a reduction of structure group to (P ∩ Q_{1})/Z_{G,P} if and only if the associated Q_{1}/Z_{G,P} Q_{2} torsor has a reduction of structure group to (P ∩Q_{1})/Z_{G,P} (P ∩Q_{2}). But Q_{1}/Q_{2} is again a semisimple, simply connected algebraic group, G′ (P ∩ Q_{1})/(P ∩ Q_{2}) is a parabolic subgroup P′, and Z_{G′,P′} equals the image of Z_{G,P}. Since the corank of P′ in G′ is 1 less than the corank of P in G, by the induction hypothesis every G′/Z_{G′,P′}-torsor over the fraction field of a surface K has a reduction of structure group to P′/Z_{G′,P′}. Thus every G/Z_{G,P} -torsor over K has a reduction of structure group to P/Z_{G,P}. Therefore the result is proved by induction on the corank. □

Proof of Theorem 1.2. This is roughly the same as the proof already presented. We include this proof to illustrate how Theorem 3.1 connects the Period-Index problem, Problem 4.15, and Serre’s Conjecture II, Conjecture 5.5. Let G be SL_{n,k}. Let m be an integer 1 < m < n and which divides n. Let P be the maximal parabolic subgroup of SL_{n,k} consisting of upper block matrices with the upper right block of size m and another diagonal block of size n − m. The center Z_{G} of SL_{n,k} is the group scheme μ_{n} of n^{th} roots of unity. And Z_{G,P} is the subgroup scheme μ_{m}.

One can prove that a torsor for G/Z_{G} = PGL_{n,k} has a reduction of structure group to G/Z_{G,P} if and only if the order of the corresponding element in the Brauer group H^{2}(Gal(K), μ_{n}) divides m. And then, by Theorem 3.1, there is a reduction of structure group to P/Z_{G,P}.

Here is a reformultation in terms of central simple algebras. Let C be a central simple algebra with center K and with dim_{K}(C) = n^{2}. Let T_{K} be the K-scheme whose set of A-points for each commutative K-algebra A equals the set of A-algebra isomorphisms of C ⊗_{K} A with Mat_{n×n,A}. Since the automorphism group of Mat_{n×n} is PGL_{n}, T_{K} is a PGL_{n,k}-torsor over K. By the previous paragraph, if the order of [C] in the Brauer group of K divides m, then there is a reduction of structure group to P/Z_{G,P}. But this is the same thing as an isomorphism of C with Mat_{m×m,K}⊗_{K}B for some central simple algebra B. In particular, if D is a division algebra over K with dim_{K}(D) = n^{2}, then D is not isomorphic to Mat_{m×m,K} ⊗_{K} B. Thus the order of [D] does not divide m. Since this holds for every proper divisor m of n, the conclusion is that the order of [D] equals n. □

Proof of Theorem 2.1. By the same argument as in the proof of Theorem 3.1, it suffices to prove the case when k has characteristic 0. Denote by B a Borel subgroup of G. Then Z_{G,B} is the trivial group scheme. So by Theorem 3.1, every G-torsor over K has a reduction of structure group to a B-torsor over K. Denote by R_{u}(B) the unipotent radical of B. Since B is connected and solvable, B/R_{u}(B) is of multiplicative type, i.e., isomorphic to G_{m,k}^{r} where r is the rank of G. By Hilbert’s Theorem 90, every torsor for B/R_{u}(B) is trivial. Thus there is a reduction of structure group to R_{u}(B). But every torsor for a unipotent group over a characteristic 0 field is trivial. Thus the B-torsor is trivial, and hence the original G-torsor was also trivial. □

Bibliography

[Art76] Michael Artin. Lectures on deformations of singularities, volume 54 of Lectures on Mathematics and Physics. Tata Institue of Fundamental Research, Bombay, 1976.

[BCTS08] M. Borovoi, J.-L. Colliot-Thélène, and A. N. Skorobogatov. The elementary obstruction and homogeneous spaces. Duke Math. J., 141(2):321–364, 2008.

[Bla91] Altha Blanchet. Function fields of generalized Brauer-Severi varieties. Comm. Algebra, 19(1):97–118, 1991.

[BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron models, volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990.

[Cam91] F. Campana. On twistor spaces of the class C. J. Differential Geom., 33(2):541–549, 1991.

[Che35] C. Chevalley. Demonstration d’une hypothèse de E. Artin. Abh. Math. Sem. Hansischen Univ., 11:73, 1935.

[CT87] Jean-Louis Colliot-Thélène. Arithmétique des variétés rationnelles et problèmes birationnels. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 641–653, Providence, RI, 1987. Amer. Math. Soc.

[CT01] J.-L. Colliot-Thélène. Die Brauersche Gruppe; ihre Verallgemeinerungen und Anwendungen in der Arithmetischen Geometrie. preprint, 2001.

[CTGP04] J.-L. Colliot-Thélène, P. Gille, and R. Parimala. Arithmetic of linear algebraic groups over 2-dimensional geometric fields. Duke Math. J., 121(2):285–341, 2004.

[CTS87] Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc. La descente sur les variétés rationnelles. II. Duke Math. J., 54(2):375–492, 1987.

[Deb01] Olivier Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001.

[Deb03] Olivier Debarre. Variétés rationnellement connexes (d’après T. Graber, J. Harris, J. Starr et A. J. de Jong). Astérisque, (290):Exp. No. 905, ix, 243–266, 2003. Séminaire Bourbaki. Vol. 2001/2002.

[dJ04] A. J. de Jong. The period-index problem for the Brauer group of an algebraic surface. Duke Math. J., 123(1):71–94, 2004.

[dJS03] A. J. de Jong and J. Starr. Every rationally connected variety over the function field of a curve has a rational point. Amer. J. Math., 125(3):567–580, 2003.

[dJS05] A. J. de Jong and J. Starr. Almost proper git-stacks and discriminant avoidance.

preprint, available http://www.math.columbia.edu/~dejong/, 2005.

[DK73] P. Deligne and N. Katz, editors. Groupes de monodromie en géométrie algébrique. II. Springer-Verlag, Berlin, 1973. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Dirigé par P. Deligne et N. Katz, Lecture Notes in Mathematics, Vol. 340.

[Esn03] Hélène Esnault. Varieties over a finite field with trivial Chow group of 0-cycles have a rational point. Invent. Math., 151(1):187–191, 2003.

[GHMS05] Tom Graber, Joe Harris, Barry Mazur, and Jason Starr. Rational connectivity and sections of families over curves. Ann. Sci. École Norm. Sup. (4), 38:671–692, 2005.

[GHS02] Tom Graber, Joe Harris, and Jason Starr. A note on Hurwitz schemes of covers of a positive genus curve. preprint, 2002.

[GHS03] Tom Graber, Joe Harris, and Jason Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1):57–67 (electronic), 2003.

[Gir71] Jean Giraud. Cohomologie non abélienne. Springer-Verlag, Berlin, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 179.

[Gro62] Alexander Grothendieck. Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.]. Secrétariat mathématique, Paris, 1962.

[Gro67] A. Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. Inst. Hautes Études Sci. Publ. Math. 20 (1964), 101-355; ibid. 24 (1965), 5-231; ibid. 28 (1966), 5-255; ibid., (32):5–361, 1967.

http://www.numdam.org/item?id=PMIHES_1965__24__5_0.

[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

[HT06] Brendan Hassett and Yuri Tschinkel. Weak approximation over function fields. Invent. Math., 163(1):171–190, 2006.

[Hur91] A. Hurwitz. Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Mathematische Annalen, 39:1–61, 1891.

[KM76] Finn Faye Knudsen and David Mumford. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand., 39(1):19–55, 1976.

[KMM92] János Kollár, Yoichi Miyaoka, and Shigefumi Mori. Rationally connected varieties. J. Algebraic Geom., 1(3):429–448, 1992.

[Kol96] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1996.

[Kol03] János Kollár. Rationally connected varieties and fundamental groups. In Higher dimensional varieties and rational points (Budapest, 2001), volume 12 of Bolyai Soc. Math. Stud., pages 69–92. Springer, Berlin, 2003.

[Laf04] Guillaume Lafon. Une surface d’Enriques sans point sur ℂ((t)). C. R. Math. Acad. Sci. Paris, 338(1):51–54, 2004.

[Lan52] Serge Lang. On quasi algebraic closure. Ann. of Math. (2), 55:373–390, 1952.

[Lie08] Max Lieblich. Twisted sheaves and the period-index problem. Compos. Math., 144(1):1–31, 2008.

[Man86] Yu. I. Manin. Cubic forms, volume 4 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1986. Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel.

[Ser79] Jean-Pierre Serre. Local fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg.

[Ser95] Jean-Pierre Serre. Cohomologie galoisienne: progrès et problèmes. Astérisque, (227):Exp. No. 783, 4, 229–257, 1995. Séminaire Bourbaki, Vol. 1993/94.

[Ser02] Jean-Pierre Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author.

[Shi74] Tetsuji Shioda. An example of unirational surfaces in characteristic p. Math. Ann., 211:233–236, 1974.

[Sko01] Alexei Skorobogatov. Torsors and rational points, volume 144 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001.

[Sta08] Jason Starr. Rational points of rationally connected and rationally simply connected varieties. preprint, 2008.

[Ste65] Robert Steinberg. Regular elements of semisimple algebraic groups. Inst. Hautes Études Sci. Publ. Math., (25):49–80, 1965.

[Sus84] A. A. Suslin. Algebraic K-theory and the norm residue homomorphism. In Current problems in mathematics, Vol. 25, Itogi Nauki i Tekhniki, pages 115–207. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.

[SVdB92] Aidan Schofield and Michel Van den Bergh. The index of a Brauer class on a Brauer-Severi variety. Trans. Amer. Math. Soc., 333(2):729–739, 1992.

[Tse33] C. Tsen. Divisionalgebren über Funktionenkörper. Nachr. Ges. Wiss. Göttingen, page 335, 1933.

[Tse36] C. Tsen. Quasi-algebraische-abgeschlossene Funktionenkörper. J. Chinese Math., 1:81–92, 1936.

[War35] E. Warning. Bemerkung zur vorstehenden Arbeit von Herrn Chevalley. Abhandl. Hamburg, 11:76–83, 1935.

[Wei56] André Weil. The field of definition of a variety. Amer. J. Math., 78:509–524, 1956.

[Wei94] Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994.