THE AMPLE CONE OF THE KONTSEVICH MODULI SPACE

IZZET COSKUN, JOE HARRIS, AND JASON STARR

Abstract. We produce ample, respectively NEF, eventually free, divisors in the Kontsevich space \overline{M}_{0,n}(ℙ^{r}, d) of n-pointed, genus 0, stable maps to ℙ^{r}, given such divisors in \overline{M}_{0,n+d}. We prove this produces all ample, respectively NEF, eventually free, divisors in \overline{M}_{0,n}(ℙ^{r}, d). As a consequence, we construct a contraction of the boundary ∪_{k=1}^{⌊d/2⌋}Δ_{k,d−k} in \overline{M}_{0,0}(ℙ^{r}, d) analogous to a contraction of the boundary ∪_{k=3}^{⌊n/2⌋}Δ̃_{k,n−k} in \overline{M}_{0,n} first constructed by Keel and M\underline{c} Kernan.

1. Introduction

Positive-dimensional families of varieties specialize – non-general varieties in the family exhibit special properties. Given a parameter space, the subset parametrizing varieties with a special property is typically closed. Which special properties occur in codimension 1, respectively for every 1-parameter family of varieties? More precisely, when is the associated closed subset an effective divisor, respectively an ample divisor? These questions, among others, motivate the study of effective and ample divisors in parameter spaces of varieties.

The parameter space we study is the Kontsevich moduli space of n-pointed, genus 0, stable maps to projective space, denoted \overline{M}_{0,n}(ℙ^{r}, d). Here we study the ample cone, and more generally the NEF and eventually free cones. In a second article [CHS], using significantly different methods and under additional hypotheses, we study the effective cone.

Our goal is to study families of curves in a general target X. Fortunately, this largely reduces to the study for ℙ^{r}: As the Kontsevich space is functorial in the target, for every morphism X → ℙ^{r}, NEF and base-point-free divisors on \overline{M}_{0,n}(ℙ^{r}, d) give NEF and base-point-free divisors on \overline{M}_{0,n}(X, β) (this functoriality is one of many advantages of the Kontsevich space over the Hilbert scheme and the Chow variety).

Here is our main result.

Theorem 1.1. Let r and d be positive integers, n a nonnegative integer such that n + d ≥ 3. There is an injective linear map,

v : Pic(\overline{M}_{0,n+d})_{ℚ}^{𝔖_{d}} → Pic(\overline{M}_{0,n}(ℙ^{r}, d))_{ℚ}.

2000 Mathematics Subject Classification. Primary:14D20, 14E99, 14H10.

During the preparation of this article the second author was partially supported by the NSF grant DMS-0200659. The third author was partially supported by the NSF grant DMS-0353692 and a Sloan Research Fellowship.

The NEF cone of \overline{M}_{0,n}(ℙ^{r}, d), respectively, the base-point-free cone, is the product of the cone generated by H, T , L_{1}, …, L_{n} and the image under v of the NEF cone of \overline{M}_{0,n+d}//𝔖_{d}, respectively, the base-point-free cone.

The action of 𝔖_{d} on \overline{M}_{0,n+d} permutes the last d marked points. The map v generates NEF and base-point-free divisors on the Kontsevich space from NEF and base-point-free divisors on \overline{M}_{0,n+d}//𝔖_{d}. In particular, it generates contractions of the Kontsevich space from contractions of \overline{M}_{0,n+d}//𝔖_{d}.

Theorem 1.2. For every integer r ≥ 1 and d ≥ 2, there is a contraction,

cont : \overline{M}_{0,0}(ℙ^{r}, d) → Y,

restricting to an open immersion on the interior M_{0,0}(ℙ^{r}, d) and whose restriction to the boundary divisor Δ_{k,d−k} ≅ M_{0,1}(ℙ^{r}, k) ×_{ℙ^{r}} M_{0,1}(ℙ^{r}, d − k) factors through the projection to \overline{M}_{0,1}(ℙ^{r}, d − k) for each 1 ≤ k ≤ ⌊d/2⌋. The following divisor is the pull-back of an ample divisor on Y,

D_{r,d} = T + ∑_{k=2}^{⌊d/2⌋} k(k − 1)Δ_{k,d−k}.

Some connection between the ample cone of the Kontsevich space and the ample cone of \overline{M}_{0,n} is natural, and certainly not surprising to experts. A similar connection between the Fulton-MacPherson space and \overline{M}_{0,n} was proved in [Che]. The primary importance of Theorem 1.1 is the precise, simple description of v: with one exception, it maps each boundary divisor of \overline{M}_{0,n+d} to the corresponding boundary divisor of the Kontsevich space. This is used to construct the contraction in Theorem 1.2, which is analogous to the “democratic” contraction of the boundary of \overline{M}_{0,n} first constructed in an unpublished note of Keel and M\underline{c} Kernan.

Recently we were informed of different constructions of the contraction of Theorem 1.2 in [Par] and by Anca Mustaţǎ and Andrei Mustaţǎ. One advantage of our proof is that it uses only the existence of the map v, which is itself a formal consequence of the definition of the Kontsevich space. The proof of Theorem 1.2 also gives a new, very short construction of Keel-M\underline{c} Kernan’s contraction of \overline{M}_{0,n}.

Acknowledgments: We would like to thank B. Hassett and A. J. de Jong for suggesting helpful references and for useful discussions.

2. Statement of results

The Kontsevich moduli space \overline{M}_{0,n}(ℙ^{r}, d) compactifies the scheme parameterizing smooth, rational curves of degree d in ℙ^{r}. Precisely, it is the smooth, proper, Deligne-Mumford stack parameterizing families of data (C, (p_{1}, ..., p_{n}), f) of,

(i) a proper, connected, at-worst-nodal, genus 0 curve C, (ii) an ordered sequence p_{1}, …, p_{n} of distinct, smooth points of C, (iii) and a degree-d morphism f : C → ℙ^{r} satisfying the following stability

condition: every irreducible component of C mapped to a point under f contains at least 3 special points, i.e., marked points p_{i} and nodes of C.

In [Pa] R. Pandharipande gives generators of the Kontsevich space:

(1) the class H of the divisor of maps whose images intersect a fixed codimension two linear space in ℙ^{r} (provided r > 1 and d > 0), (2) the class L_{i} of the pull-back ev_{i}^{∗}(O_{ℙ^{r}}(1)), for 1 ≤ i ≤ n, associated to the

i^{th} evaluation morphism, ev_{i}(C, (p_{1}, …, p_{n}), f) := f(p_{i}),

(3) and the classes Δ_{(A,d_{A}),(B,d_{B})} of the boundary divisors consisting of maps with reducible domains. Here A ⊔ B is any ordered partition of the marked points, and d_{A} and d_{B} are non-negative integers satisfying d = d_{A} + d_{B}. If d_{A} = 0 (respectively, if d_{B} = 0), we demand #A ≥ 2 (resp. #B ≥ 2).

The divisor classes H and L_{i} on \overline{M}_{0,n}(ℙ^{r}, d) are NEF and base-point-free. For d ≥ 2, there is another NEF and base-point-free divisor class T , the tangency divisor: Fixing a hyperplane Π ⊂ ℙ^{r}, T is the class of the divisor parametrizing stable maps (C, p_{1}, …, p_{i}, f) for which f^{−1}(Π) is not simply d reduced, smooth points of C. In terms of Pandharipande’s generators, the class of T equals,

T = \frac{d − 1}{d}H + ∑_{k=0}^{⌊d/2⌋} \frac{k(d − k)}{d} (∑_{A,B} Δ_{(A,k),(B,d−k)}) .

Finally, the map v from Theorem 1.1 is described in Section 3. Together, all nonnegative-linear combinations of these divisors give a cone in Pic(\overline{M}_{0,n}(ℙ^{r}, d))_{ℚ}. We use the method of test families to prove this is the entire cone of NEF divisors, respectively, eventually free divisors. In other words, we find morphisms from test varieties to \overline{M}_{0,n}(ℙ^{r}, d)_{ℚ}. Since every NEF divisor, resp. eventually free divisor, pulls back to such a NEF divisor, resp. eventually free divisor, this constrains the NEF and eventually free divisors among all divisors. By producing sufficiently many test families, we prove every NEF, resp. eventually free divisor, is in our cone.

Hypothesis 2.1. For the rest of the paper assume that the triple (n, r, d) in \overline{M}_{0,n}(ℙ^{r}, d) satisfies r ≥ 1, d ≥ 1 and n + d ≥ 3.

*^{C} ^{L}  ^{L}  ^{L}  ^{f}  ^{ℙ^{r}}^{L}^{p}

Figure 1. The morphism α.

The morphism α. There is a 1-morphism α : \overline{M}_{0,n+d} × ℙ^{r−1} → \overline{M}_{0,n}(ℙ^{r}, d) defined as follows. Fix a point p ∈ ℙ^{r} and a line L ⊂ ℙ^{r} containing p. To every curve C in \overline{M}_{0,n+d} attach a copy of L at each of the last d marked points and denote the resulting curve by C′. Consider the morphism f : C′ → ℙ^{r} that contracts C to p and maps the d rational tails isomorphically to L (see Figure 1). Since the space of lines in ℙ^{r} passing through the point p is parameterized by ℙ^{r−1}, there is an induced 1-morphism α : \overline{M}_{0,n+d} × ℙ^{r−1} → \overline{M}_{0,n}(ℙ^{r}, d).

Since α is invariant for the action of 𝔖_{d} permuting the last d marked points, the pull-back map determines a homomorphism

α^{∗} = (α_{1}^{∗}, α_{2}^{∗}) : Pic(\overline{M}_{0,n}(ℙ^{r}, d)) → Pic(\overline{M}_{0,n+d})^{𝔖_{d}} × Pic(ℙ^{r−1}).

We will denote the two projections of α^{∗} by α_{1}^{∗} and α_{2}^{∗}.

L * C p_{i} slide p^{i} along L

Figure 2. The morphism β_{i}.

The morphisms β_{i}. For each 1 ≤ i ≤ n, there is a 1-morphism β_{i} : ℙ^{1} → \overline{M}_{0,n}(ℙ^{r}, d) defined as follows. Fix a degree-(d − 1), (n − 1)-pointed curve C containing all except the i^{th} marked point. At a general point of C, attach a line L. The resulting degree-d, reducible curve will be the domain of our map. The final, i^{th} marked point is in L. Varying p_{i} in L gives a 1-morphism β_{i} : ℙ^{1} → \overline{M}_{0,n}(ℙ^{r}, d) (see Figure 2). This definition has to be slightly modified in the cases (n, d) = (1, 1) or (2, 1). When (n, d) = (1, 1), we assume that the line L with the varying marked point p_{i} constitutes the entire stable map. When (n, d) = (2, 1), we assume that the map has L as the only component. One marked point is allowed to vary on L and the remaining marked point is held fixed at a point p ∈ L.

C  L ℙ^{r} C L  L  f slide attachment point

Figure 3. The morphism γ.

The morphism γ. If d ≥ 2, there is a 1-morphism γ : ℙ^{1} → \overline{M}_{0,n}(ℙ^{r}, d) defined as follows. Take two copies of a fixed line L attached to each other at a variable point. Fix a point p in the second copy of L. Let C be a smooth, degree- (d − 2), genus 0, (n + 1)-pointed stable map to ℙ^{r} whose (n + 1)-st point maps to p. Attach this to the second copy of L at p. Altogether, this gives a degree-d, n-pointed, genus 0 stable maps with three irreducible components. The n marked points are the first n marked points of C. The only varying aspect of this family of stable maps is the attachment point of the two copies of L. Varying the attachment point in L ≅ ℙ^{1} gives a stable map parameterized by ℙ^{1}, hence there is an induced 1-morphism γ : ℙ^{1} → \overline{M}_{0,n}(ℙ^{r}, d) (see Figure 3). When (n, d) = (1, 2), we modify the definition by assuming that the map consists only of the two copies of the line L and the marked point is held fixed at the point p on the second copy of L.

Notation 2.2. If d ≥ 2, denote by P_{r,n,d} the Abelian group

P_{r,n,d} := Pic(\overline{M}_{0,n+d})^{𝔖_{d}} × Pic(ℙ^{r−1}) × Pic(ℙ^{1})^{n} × Pic(ℙ^{1}).

Denote by u = u_{r,n,d} : Pic(\overline{M}_{0,n}(ℙ^{r}, d)) → P_{r,n,d} the pull-back map

u_{r,n,d} = (α^{∗}, (β_{1}^{∗}, …, β_{n}^{∗}), γ^{∗}).

If d = 1, denote by P_{r,n,1} the Abelian group

P_{r,n,1} := Pic(\overline{M}_{0,n+d})^{𝔖_{d}} × Pic(ℙ^{r−1}) × Pic(ℙ^{1})^{n}

and denote by u = u_{r,n,1} : Pic(\overline{M}_{0,n}(ℙ^{r}, 1)) → P_{r,n,1} the pull-back map

u_{r,n,1} = (α^{∗}, (β_{1},^{∗} , …, β_{n}^{∗}))

Theorem 1.1 is equivalent to the following.

Theorem 2.3. The map u_{r,n,d} ⊗ℚ : Pic(\overline{M}_{0,n}(ℙ^{r}, d))_{ℚ} → P_{r,n,d} ⊗ℚ is an isomorphism. The image under u_{r,n,d} ⊗ ℚ of the ample cone, resp. NEF, eventually free cone of \overline{M}_{0,n}(ℙ^{r}, d) equals the product of the ample cones, resp. NEF, eventually free cones of Pic(\overline{M}_{0,n+d})^{𝔖_{d}}, Pic(ℙ^{r−1}), and the factors Pic(ℙ^{1}).

This is equivalent to Theorem 1.1 because the linear map u_{r,n,d} is simply the inverse of the product of the linear map v and the maps ℚ → Pic(\overline{M}_{0,n}(ℙ^{r}, d))_{ℚ} associated to each generator H, T and L_{1}, …, L_{n}.

Notation 2.4. Denote by \underline{n} the set \{1, …, n\}. Denote by △ = △_{n,d} the set of 4-tuples ((A, d_{A}), (B, d_{B})) of an ordered partition A ⊔ B of \underline{n} and an ordered pair of nonnegative integers (d_{A}, d_{B}) such that d_{A} + d_{B} = d and #A ≥ 2 (#B ≥ 2) if d_{A} = 0 (d_{B} = 0, respectively). Denote by △′ the subset of △ of data such that #A + d_{A} ≥ 2 and #B + d_{B} ≥ 2.

Recall that the group Pic(\overline{M}_{0,n}) is generated by boundary divisors Δ̃_{A,B}, where A ⊔ B is an ordered partition of \underline{n} with #A ≥ 2 and #B ≥ 2. Let Δ̃_{k,n−k} denote the sum of the boundary divisors ∑_{(A,B)} Δ̃_{A,B}, where the sum runs over pairs (A, B) such that #A = k and #B = n − k. The group Pic(\overline{M}_{0,n+d})^{𝔖_{d}} is generated by boundary divisors Δ̃_{(A,d_{A}),(B,d_{B})}, where ((A, d_{A}), (B, d_{B})) ∈ Δ′. The divisor Δ̃_{(A,d_{A}),(B,d_{B})} denotes the 𝔖_{d}-invariant sum of boundary divisors ∑_{(A′,B′)} Δ̃_{(A,A′),(B,B′)}, where the sum runs over pairs (A′, B′) such that A′ ⊔ B′ is a partition of the last d points and #A′ = d_{A} and #B′ = d_{B}.

To apply Theorem 2.3, we need to express the images of the standard generators of Pic(\overline{M}_{0,n}(ℙ^{r}, d)) in terms of the standard generators for Pic(\overline{M}_{0,n+d})^{𝔖_{d}}, Pic(ℙ^{r−1}) and Pic(ℙ^{1}) factors.

Proposition 2.5. (i) Assume d ≥ 2 so that γ is defined. Then

γ^{∗}T = O_{ℙ^{1}}(2), γ^{∗}H = 0, γ^{∗}L_{i} = 0, for 1 ≤ i ≤ n.

The pull-back γ^{∗}Δ_{(A,d_{A}),(B,d_{B})} = 0 unless (#A, d_{A}) or (#B, d_{B}) is equal to (0, 1) or (0, 2). Moreover, if (n, d) ≠ (0, 3),

γ^{∗}Δ_{(∅,1),(\underline{n},d−1)} = O_{ℙ^{1}}(4) and γ^{∗}Δ_{(∅,2),(\underline{n},d−2)} = O_{ℙ^{1}}(−1).

If (n, d) = (0, 3), then γ^{∗}Δ_{(∅,1),(\underline{n},d−1)} = O_{ℙ^{1}}(3).

(ii) Assume n ≥ 1 so that β_{1}, …, β_{n} are defined. Then

β_{i}^{∗}H = 0, β_{i}^{∗}L_{i} = O_{ℙ^{1}}(1), β_{i}^{∗}L_{j} = 0 if j ≠ i, and β_{i}^{∗}T = 0.

For every 1 ≤ i ≤ n, the pull-back β_{i}^{∗}Δ_{(∅,1),(\underline{n},d−1)} equals O_{ℙ^{1}}(1) if (n, d) ≠ (1, 2), and equals O_{ℙ^{1}}(2) if (n, d) = (1, 2). If (n, d) ≠ (1, 2), then β_{i}^{∗}Δ_{({i},1),({i}^{c},d−1)} equals O_{ℙ^{1}}(−1). And β_{i}^{∗}Δ_{(A,d_{A}),(B,d_{B})} equals 0 if neither (A, d_{A}) nor (B, d_{B}) equal (∅, 1) or ({i}, 1).

(iii) α^{∗}H = (0, O_{ℙ^{r−1}}(d)), α^{∗}L_{i} = 0, for 1 ≤ i ≤ n, α^{∗}T = 0.

Divisors in \overline{M}_{0,n}(ℙ^{r}, d)  α_{1}^{∗}  α_{2}^{∗}  β_{i}^{∗}  γ^{∗}
T  0  0  0  O_{ℙ^{1}}(2)
H  0  O_{ℙ^{r−1}}(d)  0  0
L_{i}  0  0  O_{ℙ^{1}}(1)  0
L_{j≠i}  0  0  0  0
Δ_{(∅,1),(\underline{n},d−1)}  c  O_{ℙ^{r−1}}(−d) O_{ℙ^{1}}(−1) O_{ℙ^{1}}(4)
Δ_{(∅,2),(\underline{n},d−2)}  Δ̃_{(∅,2),(\underline{n},d−2)}  0  0  O_{ℙ^{1}}(−1)
Δ_{({i},1),({i}^{c},d−1)}  Δ̃_{({i},1),({i}^{c},d−1)}  0  O_{ℙ^{1}}(−1)  0
Δ_{(A,d_{A}),(B,d_{B})}  Δ̃_{(A,d_{A}),(B,d_{B})}  0  0  0

all others

Figure 4. The pull-backs of the standard generators

If #A + d_{A}, #B + d_{B} ≥ 2, then α^{∗}Δ_{(A,d_{A}),(B,d_{B})} equals Δ̃_{(A,d_{A}),(B,d_{B})}. The pullback α^{∗}Δ_{(∅,1),(\underline{n},d−1)} equals (c, O_{ℙ^{r−1}}(−d)), where c is the class

c = \frac{−1}{(n + d − 1)(n + d − 2)} ∑_{{((A,d_{A}),(B,d_{B})) \atop ∈Δ′}} d_{A}(d_{B} + #B)(d_{B} + #B − 1)Δ̃_{(A,d_{A}),(B,d_{B})}.

Proof. Items (i) and (ii) follow from Lemma 3.5 and Lemma 4.1. Item (iii), except for the computation of c, is straightforward. The class c equals − ∑_{i=1}^{d} ψ_{n+i}. To rewrite this as above, use [Pa, Lemma 2.2.1] (cf. also [dJS, Lemma 6.10]). □

With the exceptions of (n, d) = (0, 3), (1, 2), and (1, 3), Proposition 2.5 is summarized by Figure 4. The phrase “all others” means, all pairs ((A, d_{A}), (B, d_{B})) such that neither ((A, d_{A}), (B, d_{B})) nor ((B, d_{B}), (A, d_{A})) already occur in the table. The lines γ^{∗} and Δ_{(∅,2),(\underline{n},d−2)} apply only if d ≥ 2. The lines L_{i}, L_{j} and Δ_{({i},1),({i}^{c},d−1)} only apply if n ≥ 1.

In [Kaw], Kawamata associated an effective, NEF ℚ-Cartier divisor L on \overline{M}_{0,n} to every n-tuple of rational numbers, (d_{1}, …, d_{n}) satisfying 0 < d_{i} ≤ 1 and d_{1} + ⋯ + d_{n} = 2. In an unpublished note Keel and M\underline{c} Kernan proved the following.

Theorem 2.6 (Keel-M\underline{c} Kernan). The ℚ-Cartier divisor L is eventually free.

In particular, when d_{1} = ⋯ = d_{n} = 2/n, the divisor class of L equals (1/n(n − 1))D_{n}, where

D_{n} = ∑_{k=2}^{⌊n/2⌋} k(k − 1)Δ̃_{k,n−k}.

This is the divisor class giving the “democratic” contraction of the boundary of \overline{M}_{0,n}, cf. [Has, §2.1.2]. One application of Theorem 2.3 is the construction of the analogous contraction in Theorem 1.2 as well as a new, short construction of the democratic contraction.

Theorem 2.7. For every integer n ≥ 4, there is a contraction

cont : \overline{M}_{0,n} → Y

restricting to an open immersion on the interior M_{0,n} and whose restriction to the boundary divisor Δ_{k,n−k} = M_{0,k+1} × M_{0,n+1−k} factors through projection to M_{0,n+1−k} for each 3 ≤ k ≤ ⌊n/2⌋. The divisor D_{n} is the pull-back of an ample divisor on Y.

It follows easily that for every rational number b satisfying

\frac{2}{(n − 1)} < b < \frac{2}{⌊n/2⌋} resp. b = \frac{2}{⌊n/2⌋},

setting B = b (cont)_{∗}(Δ̃_{2,n−2}), K_{Y} + B is an ample ℚ-Cartier divisor, and (Y, B) is Kawamata log terminal, resp. log canonical (for this one only needs the existence of the contraction and the formula for L).

3. The splitting homomorphism

In this section we define a map v : Pic(\overline{M}_{0,n+d})^{𝔖_{d}} → Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ that maps the NEF divisors in Pic(\overline{M}_{0,n+d})^{𝔖_{d}} to NEF divisors in \overline{M}_{0,n}(ℙ^{r}, d). The map v gives a splitting of the map α_{1}^{∗} defined in the introduction and is essential for the proof of Theorem 2.3.

Let Π ⊂ ℙ^{r} be a hyperplane not containing the point p used to define the morphisms α and γ. Assume that the degree d − 1 curve used to define the morphisms β_{i} is not tangent to Π, and none of the marked points on this curve are contained in Π. Finally, assume that the degree d − 2 curve used to define the morphism γ is not tangent to Π and none of the marked points are contained in Π.

Denote by M_{0,n+d}(ℙ^{r}, d) the open substack of \overline{M}_{0,n+d}(ℙ^{r}, d) parameterizing stable maps with irreducible domain. Let

ev_{n+1,...,n+d} : M_{0,n+d}(ℙ^{r}, d) → (ℙ^{r})^{d}

be the evaluation morphism associated to the last d marked point. Denote by M_{0,n+d}(ℙ^{r}, d)_{Π} the inverse image of Π^{d} and by \overline{M}_{0,n+d}(ℙ^{r}, d)_{Π} the closure of M_{0,n+d}(ℙ^{r}, d)_{Π} in \overline{M}_{0,n+d}(ℙ^{r}, d).

\overline{M}_{0,n+d}(ℙ^{r}, d)_{Π} is 𝔖_{d}-invariant under the action of 𝔖_{d} on \overline{M}_{0,n+d}(ℙ^{r}, d) permuting the last d marked points. Denote by

π : \overline{M}_{0,n+d}(ℙ^{r}, d) → \overline{M}_{0,n}(ℙ^{r}, d)

the forgetful 1-morphism that forgets the last d marked points and stabilizes the resulting family of prestable maps. This is 𝔖_{d}-invariant. Denote by

ρ : \overline{M}_{0,n+d}(ℙ^{r}, d) → \overline{M}_{0,n+d}

the 1-morphism that stabilizes the universal family of marked prestable curves over \overline{M}_{0,n+d}(ℙ^{r}, d). This is 𝔖_{d}-equivariant.

Lemma 3.1. The 1-morphism π : M_{0,n+d}(ℙ^{r}, d)_{Π} → \overline{M}_{0,n}(ℙ^{r}, d) is étale. Denoting the image by O_{Π}, the morphism π : M_{0,n+d}(ℙ^{r}, d)_{Π} → O_{Π} is an 𝔖_{d}-torsor.

Proof. Let (C, (p_{1}, …, p_{n}, q_{1}, …, q_{d}), f) be a stable map in M_{0,n+d}(ℙ^{r}, d)_{Π}. Then (C, (p_{1}, …, p_{n}), f) satisfies,

(i) C is irreducible, (ii) f^{−1}(Π) is a reduced Cartier divisor, and (iii) none of the marked points p_{i} is contained in f^{−1}(Π).

Conversely, for every stable map (C, (p_{1}, …, p_{n}), f) satisfying (i)–(iii), for every labeling of f^{−1}(Π) as q_{1}, …, q_{d}, (C, (p_{1}, …, p_{n}, q_{1}, …, q_{d}), f) is a stable map in M_{0,n+d}(ℙ^{r}, d)_{Π}. Thus O_{Π} is the open substack of stable maps satisfying (i)–(iii) and M_{0,n+d}(ℙ^{r}, d)_{Π} is the 𝔖_{d}-torsor over O_{Π} parameterizing labelings of the fibers of f^{−1}(Π). □

Denote by q : \overline{M}_{0,n+d} → \overline{M}_{0,n+d}/𝔖_{d} the geometric quotient. The composition q ∘ ρ : \overline{M}_{0,n+d}(ℙ^{r}, d)_{Π} → \overline{M}_{0,n+d}/𝔖_{d} is 𝔖_{d}-equivariant. Because M_{0,n+d}(ℙ^{r}, d)_{Π} is an 𝔖_{d}-torsor over O_{Π}, there is a unique 1-morphism ϕ′_{Π} : O_{Π} → \overline{M}_{0,n+d}/𝔖_{d} such that ϕ′ ∘ π = q ∘ ρ.

Definition 3.2. Define U_{Π} to be the maximal open substack of \overline{M}_{0,n}(ℙ^{r}, d) over which ϕ′_{Π} extends to a 1-morphism, denoted

ϕ_{Π} : U_{Π} → \overline{M}_{0,n+d}/𝔖_{d}.

Define I_{Π} to be the normalization of the closure in \overline{M}_{0,n}(ℙ^{r}, d) × \overline{M}_{0,n+d}/𝔖_{d} of the image of the graph of ϕ′_{Π}, i.e., I_{Π} is the normalization of the image of (π, q ∘ ρ). Define Ĩ_{Π} to be the normalization of the image of (π, ρ) in \overline{M}_{0,n}(ℙ^{r}, d) × \overline{M}_{0,n+d}. Finally, define Ũ_{Π} to be the inverse image of U_{Π} in Ĩ_{Π}.

There is a pull-back map of 𝔖_{d}-invariant invertible sheaves,

ρ^{∗} : Pic(\overline{M}_{0,n+d})^{𝔖_{d}} → Pic(Ĩ_{Π})^{𝔖_{d}},

which further restricts to Pic(Ũ_{Π})^{𝔖_{d}}. After étale base-change from U_{Π} to a scheme, the morphism Ũ_{Π} → U_{Π} is the geometric quotient of Ũ_{Π} by the action of 𝔖_{d}. Therefore the pull-back map Pic(U_{Π}) → Pic(Ũ_{Π})^{𝔖_{d}} is an isomorphism after tensoring with ℚ; in fact, both the kernel and cokernel are annihilated by d!. Because \overline{M}_{0,n+d}/𝔖_{d} is a proper scheme and because \overline{M}_{0,n}(ℙ^{r}, d) is separated and normal, by the valuative criterion of properness the complement of U_{Π} has codimension ≥ 2. The smoothness of \overline{M}_{0,n}(ℙ^{r}, d) and [Ha, Prop. 6.5(c)] imply that the restriction map Pic(\overline{M}_{0,n}(ℙ^{r}, d)) → Pic(U_{Π}) is an isomorphism.

Definition 3.3. Define v : Pic(\overline{M}_{0,n+d})^{𝔖_{d}} → Pic(\overline{M}_{0,n}(ℙ^{r}, d))⊗ℚ to be the unique homomorphism commuting with ρ^{∗} via the isomorphisms above.

The map v is independent of the choice of Π, hence it sends NEF divisors to NEF divisors.

Lemma 3.4. For every base-point-free invertible sheaf L in Pic(\overline{M}_{0,n+d})^{𝔖_{d}}, v(L) is base-point-free. In particular, for every ample invertible sheaf L, v(L) is NEF. Thus, by Kleiman’s criterion, for every NEF invertible sheaf L, v(L) is NEF.

Proof. For every [(C, (p_{1}, …, p_{n}), f)] in \overline{M}_{0,n}(ℙ^{r}, d), there exists a hyperplane Π satisfying the conditions above and such that f^{−1}(Π) is a reduced Cartier divisor containing none of p_{1}, …, p_{n}. By Lemma 3.1, (C, (p_{1}, …, p_{n}), f) is contained in U_{Π}. Since L is base-point-free, there exists a divisor D in the linear system |L| not containing ϕ_{Π}[(C, (p_{1}, …, p_{n}), f)]. By the proof of [Ha, Prop. 6.5(c)], the closure of ϕ_{Π}^{−1}(D) is in the linear system |v(L)|; and it does not contain [(C, (p_{1}, …, p_{n}), f)].

□

Lemma 3.5. (i) The images of α, β_{i} and γ are contained in U_{Π}.

(ii) The morphisms ϕ_{Π} ∘β_{i} and ϕ_{Π} ∘γ are constant morphisms. Therefore β_{i}^{∗} ∘v

and γ^{∗} ∘ v are the zero homomorphism. (iii) The composition of α with ϕ_{Π} equals q ∘ pr_{\overline{M}_{0,n+d}}. Therefore

α^{∗} ∘ v : Pic(\overline{M}_{0,n+d})^{𝔖_{d}} → Pic(\overline{M}_{0,n+d})^{𝔖_{d}} × Pic(ℙ^{r−1})

is the homomorphism whose projection on the first factor is the identity, and whose projection on the second factor is 0.

Proof. (i): The image of α is contained in O_{Π}. Denote by q the intersection point of L and Π.

The image β_{i}(L − {q}) is contained in O_{Π}. The stable map β_{i}(q) sends the i^{th} marked point into Π. Up to labeling the d points of the inverse image of Π, there is only one (n+d)-pointed stable map in \overline{M}_{0,n+d}(ℙ^{r}, d)_{Π} that stabilizes to this stable map. It is obtained from β_{i}(q) by removing the i^{th} marked point from L, attaching a contracted component C′ to L at q, containing the i^{th} marked point and exactly one of the last d marked points, and labeling the d − 1 points in C ∩ Π with the remaining d − 1 marked points.

Similarly, γ(L − {q}) is contained in O_{Π}. The stable map γ(q) has two copies of L attached to each other at q. This appears to be a problem, because the inverse image of γ(q) in \overline{M}_{0,n+d}(ℙ^{r}, d)_{Π} is 1-dimensional, isomorphic to \overline{M}_{0,4}. The stable maps have a contracted component C′ such that both copies of L are attached to C′ and 2 of the d new marked points are attached to C′. The remaining d − 2 marked points are the points of C ∩ Π. However, the map ρ that stabilizes the resulting prestable (n + d)-marked curve is constant on this \overline{M}_{0,4}. Indeed, the first copy of L has no marked points and is attached to C′ at one point. So the first step in stabilization will prune L reducing the number of special points on C′ from 4 to 3.

(ii): In the family defining β_{i}, only the i^{th} marked point on L varies. After adding the d new marked points, L is a 3-pointed prestable curve; marked by the node p, the i^{th} marked point, and the point q. For every base the only family of genus 0, 3-pointed, stable curves is the constant family. So upon stabilization, this family of genus 0, 3-pointed, stable curves becomes the constant family.

In the family defining γ, only the attachment point of the two copies of L varies. The first copy of L gives a family of 2-pointed, prestable curves; marked by q and the attachment point of the two copies of L. This is unstable. Upon stabilization, the first copy of L is pruned and the marked point q on the first copy is replaced by a marked point on the second copy at the original attachment point. Now the second copy of L gives a family of 3-pointed, prestable curves; marked by the attachment point p of the second and third irreducible components, the attachment point of the first and second components, and q. For the same reason as in the last paragraph, this becomes a constant family.

(iii): Each stable map in α(\overline{M}_{0,n+d} × ℙ^{r−1}) is obtained from a genus 0, (n + d)- pointed, stable curve (C_{0}, (p_{1}, …, p_{n}, q_{1}, …, q_{d})) and a line L in ℙ^{r} containing p by attaching for each 1 ≤ i ≤ n, a copy C_{i} of L to C_{0} where p in C_{i} is identified with q_{i} in C_{0}. The map to ℙ^{r} contracts C_{0} to p, and sends each curve C to L via the identity morphism. Denoting by r the intersection point of L and Π, the inverse image of Π consists of the d points r_{1}, …, r_{d}, where r_{i} is the copy of r in C_{i}.

The component C_{i} is a 2-pointed, prestable curve: marked by the attachment point p of C_{i} and by r_{i}. This is unstable. So, upon stabilization, C_{i} is pruned and the marked point r_{i} is replaced by a marking on C_{0} at the point of attachment of C_{0} and C_{i}, namely q_{i}. Therefore, up to relabeling of the last d marked points, the result is the genus 0, (n + d)-pointed, stable curve we started with, (C_{0}, (p_{1}, …, p_{n}, q_{1}, …, q_{d})). □

4. More divisors

In the previous section we constructed a map (see Definition 3.3)

v : Pic(\overline{M}_{0,n+d})^{𝔖_{d}} → Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ.

In this section we prove that the image of v, the divisor classes H, T and the tautological divisors L_{i}, generate Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ.

The divisor class H_{Λ}, [Pa, Prop. 1] is the class of stable maps whose image intersects a fixed codimension 2 linear space Λ of ℙ^{r}. This is defined to be the empty divisor if r = 1. For convenience, assume Λ is contained in Π and does not intersect L or the curves C used to define β_{i} and γ. If n ≥ 1, the divisors L_{i,Π}, i = 1, …, n, [Pa, Prop. 1] are the pull-back by ev_{i} of the Cartier divisor Π. If d ≥ 1, the last divisor is T_{Π}, [Pa, §2.3]; the divisor of stable maps (C, (p_{1}, …, p_{n}), f) such that f^{−1}(Π) is not a reduced, finite set of degree d. This is defined to be the empty divisor if d = 1. In [Pa] Pandharipande proves that H_{Λ}, L_{i,Π} and T_{Π} are irreducible Cartier divisors (when they are nonempty).

Lemma 4.1. (i) The Cartier divisors T_{Π}, L_{i,Π} and H_{Λ} are NEF.

(ii) The pull-backs α^{∗}(T_{Π}) and α^{∗}(L_{i,Π}) are zero. The pull-back α^{∗}(H_{Λ}) equals (0, O_{ℙ^{r−1}}(d)) in Pic(\overline{M}_{0,n+d})^{𝔖_{d}} × Pic(ℙ^{r−1}); if r = 1, then O_{ℙ^{r−1}}(1) is the trivial invertible sheaf.

(iii) Assume n ≥ 1 so that β_{i} is defined for 1 ≤ i ≤ n. The pull-backs β_{i}^{∗}(T_{Π}) and β_{i}^{∗}(H_{Π}) are zero. For 1 ≤ j ≤ n different from i, β_{i}^{∗}(L_{j,Π}) is zero. Finally, β_{i}^{∗}(L_{i,Π}) is O_{ℙ^{1}}(1).

(iv) Assume d ≥ 2 so that γ is defined. The pull-backs γ^{∗}(H_{Λ}) and γ^{∗}(L_{i,Π}) are zero, and γ^{∗}(T_{Π}) is O_{ℙ^{1}}(2) in Pic(ℙ^{1}).

Proof. (i): By an argument similar to the one in Lemma 3.4, these divisors are base-point-free (whenever they are non-empty). The divisor H_{Λ} is big if r ≥ 2, and T_{Π} is big if d ≥ 2. The divisors L_{i} are not big.

(ii): By the proof of Lemma 3.5, the image of α is in O_{Π}, which is disjoint from T_{Π}. Also, ev_{i} ∘ α is the constant morphism with image p, so the inverse image of L_{i} is empty. Finally, the pull-back of H_{Π} equals the pull-back under the diagonal Δ of the Cartier divisor ∑_{j=1}^{d} pr_{j}^{−1}(Λ) in (ℙ^{r−1}), where Λ is considered as a divisor in ℙ^{r−1} via projection from p.

(iii): Since the image of β_{i} is disjoint from H_{Π}, T_{Π} and L_{j,Π} for j ≠ i, the corresponding pull-backs are zero. The map ev_{i} ∘ β_{i} : ℙ^{1} → ℙ^{r} embeds ℙ^{1} as the line L in ℙ^{r}, hence β_{i}^{∗}(L_{i,Π}) = O_{ℙ^{1}}(1).

(iv): Since neither the image curve nor the marked points vary under γ, clearly γ^{∗}H_{Λ} and γ^{∗}L_{i,Π} are zero. To compute γ^{∗}T_{Π}, use [Pa, Lem 2.3.1]. □

The main observation of this section is the following.

Proposition 4.2. The ℚ-vector space Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ is generated by T_{Π}, H_{Λ}, L_{i,Π} for 1 ≤ i ≤ n, and the image of v.

Proof. When r ≥ 2, Pandharipande proves that the classes of the divisors H_{Λ}, L_{i,Π} for 1 ≤ i ≤ n, and the boundary divisors Δ_{(A,d_{A}),(B,d_{B})} for ((A, d_{A}), (B, d_{B})) ∈ Δ generate the ℚ-vector space Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ, cf. [Pa, Prop. 1]. The tangency divisor T can be expressed in terms of H and the boundary divisors as follows [Pa, Lem 2.3.1]:

T = \frac{d − 1}{d}H + ∑_{j=0}^{⌊ \frac{d}{2} ⌋} \frac{j(d − j)}{d} ∑_{((A,d_{A}),(B,d_{B})),d_{A}=j} Δ_{(A,d_{A}),(B,d_{B})}.

From Lemmas 4.1 and 3.5 and by pairing with one-parameter families, we see that

v(Δ̃_{(A,d_{A}),(B,d_{B})}) = Δ_{(A,d_{A}),(B,d_{B})}

unless (#A, d_{A}) or (#B, d_{B}) equals one of (0, 2) or (1, 1).

v(Δ̃_{(A,d_{A}),(B,d_{B})}) = \frac{1}{2} T + Δ_{(A,d_{A}),(B,d_{B})}

if (#A, d_{A}) or (#B, d_{B}) equals (0, 2). Finally,

v(Δ̃_{({i},1),({i}^{c},d−1)}) = Δ_{({i},1),({i}^{c},d−1)} + L_{i,Π}.

Consequently, it follows that the classes of the divisors H, T , L_{i,Π} and the image of v generate the classes of all the boundary divisors in the Kontsevich moduli space. Hence, they generate Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ.

We can reduce the case r = 1 to the case r ≥ 2. Because L is disjoint from Λ, there is a unique linear projection

pr_{Λ} : (ℙ^{r} − Λ) → L

whose restriction to L is the identity. This is a vector bundle over L whose associated sheaf of sections is O_{L}(1)^{⊕(r−1)}. Composing a stable map to (ℙ^{r} − Λ) with pr_{Λ} gives a stable map to L. This defines a 1-morphism,

\overline{M}_{0,n}(pr_{Λ}, d) : (\overline{M}_{0,n}(ℙ^{r}, d) − H_{Λ}) → \overline{M}_{0,n}(L, d).

This is a vector bundle over \overline{M}_{0,n}(L, d) whose associated sheaf of sections is the sheaf whose fiber at (C, (p_{1}, …, p_{n}), f) equals H^{0}(C, f^{∗}O_{L}(1)^{⊕(r−1)}). Thus the pull-back homomorphism,

\overline{M}_{0,n}(pr_{Λ}, d)^{∗} : Pic(\overline{M}_{0,n}(L, d)) → Pic(\overline{M}_{0,n}(ℙ^{r}, d) − H_{Λ}),

is an isomorphism, cf. [Ful, Thm. 3.3(a)].

The hyperplane Π is the closure of pr_{Λ}^{−1}(L ∩ Π). Thus U_{Π} − H_{Λ} ∩ U_{Π} (see Definition 3.2) is the inverse image of the corresponding open substack of \overline{M}_{0,n}(L, d) for L ∩ Π inside L. The inverse image of T_{L∩Π}, resp. L_{i,L∩Π}, equals the restriction of T_{Π}, resp. L_{i,Π}. And ϕ_{L∩Π} ∘ \overline{M}_{0,n}(pr_{Λ}, d) equals the restriction of ϕ_{Π}. Thus Pic(\overline{M}_{0,n}(ℙ^{r}, d) − H_{Λ}) ⊗ ℚ is generated by T_{Π}, L_{i,Π} for 1 ≤ i ≤ n, and the image of v if and only if the same is true for Pic(\overline{M}_{0,n}(ℙ^{1}, d)) ⊗ ℚ.

□

5. Proof of the main theorem

In this section we complete the proof of Theorem 2.3. Recall that Theorem 2.3 asserts that the NEF cone of the Kontsevich moduli space \overline{M}_{0,n}(ℙ^{r}, d) equals the NEF cone in P_{r,n,d} ⊗ ℚ, where P_{r,n,d} is the Abelian group

P_{r,n,d} := Pic(\overline{M}_{0,n+d})^{𝔖_{d}} × Pic(ℙ^{r−1}) × Pic(ℙ^{1})^{n} × Pic(ℙ^{1}).

The identification of Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ with P_{r,n,d} ⊗ ℚ is given by the map

u = u_{r,n,d} := (α^{∗}, (β_{1}^{∗}, …, β_{n}^{∗}), γ^{∗})

(see §1).

Denote by

ṽ: P_{r,n,d} ⊗ ℚ → Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ

the unique homomorphism whose restriction to Pic(\overline{M}_{0,n+d})^{𝔖_{d}} is v (see Definition 3.3) , whose restriction to Pic(ℙ^{r−1}) sends O_{ℙ^{r−1}}(1) to [H_{Λ}], whose restriction to the i^{th} factor of Pic(ℙ^{1})^{n} sends O_{ℙ^{1}}(1) to [L_{i}] if n ≥ 1, and whose restriction to the last factor Pic(ℙ^{1}) (assuming d ≥ 2) sends O_{ℙ^{1}}(1) to 1/2 [T_{Π}]. By Lemma 3.5 (ii), (iii) and by Lemma 4.1, u ⊗ ℚ ∘ ṽ is the identity map. In particular, ṽ is injective. By Proposition 4.2, ṽ is surjective. Thus ṽ and u ⊗ ℚ are isomorphisms.

Because α, β_{i} and γ are morphisms, for every NEF, resp. eventually free, divisor D in Pic(\overline{M}_{0,n}(ℙ^{r}, d)) ⊗ ℚ, α^{∗}(D), β_{i}^{∗}(D), and γ^{∗}(D) are NEF, resp. eventually free. Denote,

D_{1} = α_{1}^{∗}(D), a [O_{ℙ^{r−1}}(1)] = α_{2}^{∗}(D), b_{i} [O_{ℙ^{1}}(1)] = β_{i}^{∗}(D), c [O_{ℙ^{1}}(1)] = γ^{∗}(D),

where by convention a is defined to be 0 if r = 1 and c is defined to be 0 if d = 1. If D is NEF, resp. eventually free, D_{1} is NEF, resp. eventually free, in Pic(\overline{M}_{0,n+d})^{𝔖_{d}}, and a, b_{i}, c ≥ 0.

Conversely, by Lemma 3.4, for every NEF, resp. eventually free, divisor D_{1} in Pic(\overline{M}_{0,n+d})^{𝔖_{d}}, v(D_{1}) is NEF, resp. eventually free. By Lemma 4.1(i), for a, b_{i}, c ≥ 0, a[H_{Λ}], b_{i}[L_{i,Π}] and c/2 [T_{Π}] are NEF and eventually free. Since a sum of NEF, resp. eventually free, divisors is NEF, resp. eventually free, D = v(D_{1}) + a [H_{Λ}] + b_{i} [L_{i}] + c/2 [T_{Π}] is NEF, resp. eventually free. Therefore D is NEF if and only if u⊗ℚ(D) is in the product of the NEF cones of the factors. This argument needs to be modified in the obvious way when (n, d) = (0, 3) and (1, 2) to account for the slight variations in the formulae.

Because the interior of a product of cones equals the product of the interiors of the cones, by Kleiman’s criterion, D is ample iff u ⊗ ℚ(D) is contained in the product of the ample cones of the factors. □

Remark 5.1. Since the analogue of the F-conjecture is known for \overline{M}_{0,d}/𝔖_{d} when d ≤ 11 by [KM], Theorem 2.3 provides an explicit description of the NEF cone of \overline{M}_{0,0}(ℙ^{r}, d) for r ≥ 2 and d ≤ 11. For example, when d = 2, 3, the NEF cone is bounded by the rays H and T . When d = 4, 5, the NEF cone is generated by the rays H, T and H + Δ_{1,d−1} + 4 Δ_{2,d−2}.

6. The contractions

Theorem 2.7(i)–(v) and Theorem 1.2 are proved simultaneously by induction on n, resp. d, in the following 2 lemmas. Since the divisor D_{n} = ∑_{k=2}^{⌊n/2⌋} k(k−1)Δ̃_{k,n−k} is ample on \overline{M}_{0,n} for n = 4, 5, the base cases n = 4, 5 for Theorem 2.7 are immediate. The base cases d = 2, 3 for Theorem 1.2 are also straightforward.

Lemma 6.1. Let d ≥ 4 be an integer. The existence of a contraction as in Theorem 2.7 for n = d implies the existence of a contraction as in Theorem 1.2 for d.

Proof. The divisor D_{r,d} = T +∑_{k=2}^{⌊d/2⌋} k(k −1)Δ_{k,d−k} equals v(D_{d}). By hypothesis, v(D_{d}) is eventually free, thus D_{r,d} is eventually free by Lemma 3.4. Define cont_{r,d} : \overline{M}_{0,0}(ℙ^{r}, d) → Y_{r,d} to be the associated morphism with connected fibers and normal target.

Denote by O_{r,d} the maximal open subscheme of Y_{r,d} over which cont_{r,d} is finite. The claim is that cont_{r,d}^{−1}(O_{r,d}) contains M_{0,0}(ℙ^{r}, d). Every proper, irreducible curve B in \overline{M}_{0,0}(ℙ^{r}, d) contracted by cont_{r,d} has intersection number 0 with D_{r,d}. If B intersects M_{0,0}(ℙ^{r}, d), the intersection number with every Δ_{k,d−k}, k = 1, …, ⌊d/2⌋ is nonnegative. Since T is NEF, the intersection number of B with T is nonnegative. Since D_{r,d} · B = 0, B has intersection number zero with T and Δ_{k,d−k}, k = 2, …, ⌊d/2⌋. From the expression of T and the fact that H and Δ_{1,d−1} have nonnegative intersection with B, it follows that B has intersection number zero with H and Δ_{1,d−1}, as well. Since there exists an ample linear combination of these divisors, we obtain a contradiction. Thus B is contained in the complement of M_{0,0}(ℙ^{r}, d), proving the claim.

By Zariski’s Main Theorem, cont_{r,d} : cont_{r,d}^{−1}(O_{r,d}) → O_{r,d} is an isomorphism. In particular, cont_{r,d} : M_{0,0}(ℙ^{r}, d) → O_{r,d} is an open immersion.

The 1-morphism ϕ_{Π} from Definition 3.2 maps Δ_{k,d−k} to Δ̃_{k,d−k} compatible with the boundary maps. Thus cont_{r,d} satisfies the conclusion of Theorem 1.2. □

Lemma 6.2. Let n ≥ 6 be an integer. The existence of a contraction as in Theorem 1.2 for d = n − 2 implies the existence of a contraction as in Theorem 2.7 for n.

Proof. Denote by ev : \overline{M}_{0,n}(ℙ^{n−2}, n − 2) → (ℙ^{n−2})^{n} the evaluation 1-morphism. Denote by Φ : \overline{M}_{0,n}(ℙ^{n−2}, n − 2) → \overline{M}_{0,n} the forgetful 1-morphism. Denote by U ⊂ (ℙ^{n−2})^{n} the open subset parameterizing n-tuples of points in linear general position, i.e., the span of every (n − 1)-tuple equals ℙ^{n−2}. Kapranov proves that the 1-morphism,

(ev, Φ) : \overline{M}_{0,n}(ℙ^{n−2}, n − 2) → (ℙ^{n−2})^{n} × \overline{M}_{0,n},

is an isomorphism over U × \overline{M}_{0,n}, [Kap]. Fix a general point q in U, and identify \overline{M}_{0,n} with the fiber over \{q\} × \overline{M}_{0,n}.

The forgetful 1-morphism π : \overline{M}_{0,n}(ℙ^{n−2}, n − 2) → \overline{M}_{0,0}(ℙ^{n−2}, n − 2) restricts to a 1-morphism p : \overline{M}_{0,n} → \overline{M}_{0,0}(ℙ^{n−2}, n − 2). Denote by cont : \overline{M}_{0,n} → Y the Stein factorization of,

cont_{n−2,n−2} ∘ p : \overline{M}_{0,n} → cont_{n−2,n−2}(p(\overline{M}_{0,n})).

It is straightforward that p^{−1}(Δ_{k−1,n−1−k}) = Δ̃_{k,n−k} for every 2 ≤ k ≤ ⌊n/2⌋ compatibly with the boundary maps. Thus cont satisfies the conclusion of Theorem 2.7. □

Remark 6.3. Pairing with test curves gives that

v (\frac{1}{n − 1}D_{n}) = \frac{1}{n − 1}D_{r,n} and p^{∗} (\frac{1}{n − 3}D_{n−2,n−2}) = \frac{1}{n − 1}D_{n}.

The image of the ample cone of \overline{M}_{0,0}(ℙ^{n}, n) under p^{∗} is not all of the ample cone of \overline{M}_{0,n+2}, already for n = 6.

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[KM] S. Keel and J. McKernan. Contractible extremal rays on \overline{M}_{0,n}. preprint, 1996.

[Pa] R. Pandharipande. Intersections of Q-divisors on Kontsevich’s moduli space \overline{M}_{0,n}(P^{r}, d) and enumerative geometry. Trans. Amer. Math. Soc. 351(1999), 1481–1505.

[Par] A. Parker. An elementary GIT construction of the moduli space of stable maps. PhD thesis, University of Texas at Austin, 2005.

Department of Mathematics, M.I.T., Cambridge, MA 02139

   E-mail address: coskun@math.mit.edu

Department of Mathematics, Harvard University, Cambridge, MA 02138

   E-mail address: harris@math.harvard.edu

Department of Mathematics, M.I.T., Cambridge, MA 02139

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