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\title[Working Draft]{The ample cone of the Kontsevich moduli space}
\author[Coskun]{Izzet Coskun}\address{Department of Mathematics,
M.I.T., Cambridge, MA 02139}
\email{coskun@math.mit.edu} \author[Harris]{Joe
Harris}\address{Department of Mathematics, Harvard University,
Cambridge, MA 02138} \email{harris@math.harvard.edu}
\author[Starr]{Jason Starr} \address{Department of Mathematics,
M.I.T., Cambridge, MA 02139}
\email{jstarr@math.mit.edu}
%\date{Fall 2004}
\subjclass[2000]{Primary:14D20, 14E99, 14H10} \thanks{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.}

\begin{document}

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\begin{abstract}
  We produce ample, respectively NEF, eventually free, divisors in the
  Kontsevich space $\Kgnb{0,n} (\PP^r, d)$ of $n$-pointed,
  genus $0$, stable maps to $\PP^r$, given such divisors in
  $\Kgnb{0,n+d}$.  We prove this produces all ample, respectively NEF,
  eventually free, divisors in $\Kgnb{0,n}(\PP^r,d)$.  
  As a consequence, we construct a contraction of the boundary
  $\cup_{k=1}^{\lfloor d/2 \rfloor} \Delta_{k,d-k}$ in
  $\Kgnb{0,0}(\PP^r,d)$ analogous to a contraction of
  the boundary $\cup_{k=3}^{\lfloor n/2 \rfloor}
  \tilde{\Delta}_{k,n-k}$ in $\kgnb{0,n}$ first constructed by Keel
  and \mac~Kernan.
\end{abstract}

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\section{Introduction} \label{sec-intro}

Positive-dimensional families of varieties \emph{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. \medskip

The parameter space we study is the Kontsevich moduli space of
$n$-pointed, genus $0$, stable maps to projective space, denoted
$\Kgnb{0,n}(\PP^r,d)$.  Here we study the ample cone, and more
generally the NEF and eventually free cones.  In a second article
\cite{coskun:effective}, using significantly different methods and
under additional hypotheses, we study the effective cone.  \medskip

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

Here is our main result.

\begin{thm} \label{thm-mainpre}
Let $r$ and $d$ be positive integers, $n$ a nonnegative integer such
that $n+d \geq 3$.  There is an injective linear map,
$$
v:\text{Pic}(\Kgnb{0,n+d})^{\mf{S}_d}_\QQ \rightarrow
\text{Pic}(\Kgnb{0,n}(\PP^r,d))_\QQ.
$$
The NEF cone of $\Kgnb{0,n}(\PP^r,d)$, respectively, the base-point-free
cone, is the product of the cone generated by $\mc{H}, \mc{T},
\mc{L}_1,\dots,\mc{L}_n$ and the
image under $v$ of the NEF cone of
$\Kgnb{0,n+d}//\mf{S}_d$, respectively, the base-point-free cone.
\end{thm}

The action of $\mf{S}_d$ on $\Kgnb{0,n+d}$ 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
$\Kgnb{0,n+d}//\mf{S}_d$.  In particular, it generates contractions of
the Kontsevich space from contractions of $\Kgnb{0,n+d}//\mf{S}_d$.

\begin{thm} \label{thm-mainpre2}
For every integer $r\geq 1$ and $d\geq 2$, there is a contraction,
$$
\text{cont}:\kgnb{0,0}(\PP^r,d) \rightarrow Y,
$$
restricting to an open immersion on the interior $\kgn{0,0}(\PP^r,d)$
and whose restriction to the boundary divisor $\Delta_{k,d-k} \cong
\kgn{0,1}(\PP^r,k)\times_{\PP^r} \kgn{0,1}(\PP^r,d-k)$ factors through
the projection to $\kgnb{0,1}(\PP^r,d-k)$ for each $1\leq k\leq
\lfloor d/2 \rfloor$.  The following divisor is the pull-back of an
ample divisor on $Y$,
$$
D_{r,d} = \mc{T} + \sum_{k=2}^{\lfloor d/2 \rfloor}
  k(k-1)\Delta_{k,d-k}.
$$
\end{thm}

Some connection between the ample cone of the Kontsevich space and the
ample cone of $\Kgnb{0,n}$ is natural, and certainly not surprising to
experts.  A similar connection between the Fulton-MacPherson space and
$\Kgnb{0,n}$ was proved in ~\cite{LChen}.  The primary importance of
Theorem~\ref{thm-mainpre} is the precise, simple description of $v$:
with one exception, it maps each boundary divisor of $\Kgnb{0,n+d}$ to
the corresponding boundary divisor of the Kontsevich space.  This is
used to construct the contraction in Theorem~\ref{thm-mainpre2}, which
is analogous to the ``democratic'' contraction of the boundary of
$\Kgnb{0,n}$ first constructed in an unpublished note of Keel and
\mac~Kernan.  \medskip

Recently we were informed of different constructions of the
contraction of Theorem~\ref{thm-mainpre2} in ~\cite{Parker} and by
Anca Musta\c{t}\v{a} and Andrei Musta\c{t}\v{a}.  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~\ref{thm-mainpre2} also gives a new, very short
construction of Keel-\mac~Kernan's contraction of $\Kgnb{0,n}$.


\medskip

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

\section{Statement of results} \label{sec-sor}

The Kontsevich moduli space $\Kgnb{0,n}(\PP^r,d)$ compactifies the
scheme parameterizing smooth, rational curves of degree $d$ in
$\PP^r$.  Precisely, it is the smooth, proper, Deligne-Mumford stack
parameterizing families of data $(C,(p_1,...,p_n),f)$ of,
\begin{enumerate}
\item[(i)]
  a proper, connected, at-worst-nodal, genus $0$ curve $C$, 
\item[(ii)]  
  an ordered sequence $p_1,\dots,p_n$ of distinct, smooth points of
  $C$,
\item[(iii)]  
  and a degree-$d$ morphism $f:C\rightarrow \PP^r$
  satisfying the following stability condition:
  every irreducible component of $C$ mapped to a point under $f$
  contains at least $3$ \emph{special points}, i.e., marked points $p_i$ and
  nodes of $C$.
\end{enumerate}

In ~\cite{pandh:qdivisors} R. Pandharipande gives generators of the
Kontsevich space:
\begin{enumerate}
\item the class $\mc{H}$ of the divisor of maps whose images intersect
  a fixed codimension two linear space in $\PP^r$ (provided $r>1$ and
  $d > 0$),

\item the class $\mc{L}_i$ of the pull-back
  $\text{ev}_i^*(\mathcal{O}_{\PP^r}(1))$, for $1 \leq i \leq n$,
  associated to the $i^\text{th}$ evaluation morphism,
  $\text{ev}_i(C,(p_1,\dots, p_n),f) := f(p_i)$,

\item and the classes $\Delta_{(A,d_A),(B,d_B)}$ of the boundary
  divisors consisting of maps with reducible domains. Here $A \sqcup
  B$ 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 $\# A \geq 2$ (resp.
  $\# B \geq 2$).
\end{enumerate}
\medskip

The divisor classes $\mc{H}$ and $\mc{L}_i$ on $\Kgnb{0,n}(\PP^r,d)$
are NEF and base-point-free.  For $d \geq 2$, there is another NEF and
base-point-free divisor class $\mc{T}$, the \emph{tangency divisor}:
Fixing a hyperplane $\Pi \subset \PP^r$, $\mc{T}$ is the class of the
divisor parametrizing stable maps $(C, p_1, \dots, p_i, f)$ for which
$f^{-1} (\Pi)$ is not simply $d$ reduced, smooth points of $C$.  In
terms of Pandharipande's generators, the class of $\mc{T}$ equals,
$$
\mc{T} = \frac{d-1}{d} \mc{H} + \sum_{k=0}^{\lfloor d/2 \rfloor}
\frac{k(d-k)}{d} \left( \sum_{A,B} \Delta_{(A,k),(B,d-k)} \right).
$$

Finally, the map $v$ from Theorem~\ref{thm-mainpre} is described in
Section ~\ref{sec-split}.  Together, all nonnegative-linear
combinations of these divisors give a cone in
$\text{Pic}(\Kgnb{0,n}(\PP^r,d))_\QQ$.  We use the method of
\emph{test families} to prove this is the entire cone of NEF divisors,
respectively, eventually free divisors.  In other words, we find
morphisms from \emph{test varieties} to $\Kgnb{0,n}(\PP^r,d)_\QQ$.
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.

\begin{hyp} \label{hyp-rnd} For the rest of the paper assume that the
  triple $(n,r,d)$ in $\Kgnb{0,n}(\PP^r,d)$ satisfies $r\geq 1, \
  d\geq 1 \ \mbox{ and} \ n+d \geq 3.$
\end{hyp}

\begin{figure}[htbp]
\begin{center}
\input{mora.pstex_t}
\end{center}
\caption{The morphism $\alpha$.}
\label{mora}
\end{figure}

{\bf The morphism $\alpha$.} There is a 1-morphism $\alpha:
\overline{M}_{0,n+d} \times \PP^{r-1} \rightarrow \Kgnb{0,n}(\PP^r,d)$
defined as follows. Fix a point $p \in \PP^r$ and a line $L \subset
\PP^r$ 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 $f: C' \rightarrow
\PP^r$ that contracts $C$ to $p$ and maps the $d$ rational tails
isomorphically to $L$ (see Figure \ref{mora}). Since the space of
lines in $\PP^r$ passing through the point $p$ is parameterized by
$\PP^{r-1}$, there is an induced 1-morphism $\alpha:
\overline{M}_{0,n+d} \times \PP^{r-1} \rightarrow
\Kgnb{0,n}(\PP^r,d).$

Since $\alpha$ is invariant for the action of $\mathfrak{S}_d$
permuting the last $d$ marked points, the pull-back map determines a
homomorphism
$$
\alpha^* = (\alpha_1^*,\alpha_2^*): \text{Pic}(\Kgnb{0,n}(\PP^r,d))
\rightarrow \text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \times
\text{Pic}(\PP^{r-1}).
$$ 
We will denote the two projections of $\alpha^*$ by $\alpha_1^*$
and $\alpha_2^*$.\smallskip

\begin{figure}[htbp]
\begin{center}
\input{morb.pstex_t}
\end{center}
\caption{The morphism $\beta_i$.}
\label{morb}
\end{figure}

{\bf The morphisms $\beta_i$.} For each $1 \leq i \leq n$, there is a
1-morphism $\beta_i : \PP^1 \rightarrow \Kgnb{0,n}(\PP^r,d)$ defined
as follows. Fix a degree-$(d-1)$, $(n-1)$-pointed curve $C$ containing
all except the $i^\text{th}$ 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, $i^\text{th}$ marked point is in
$L$.  Varying $p_i$ in $L$ gives a 1-morphism $\beta_i: \PP^1
\rightarrow \Kgnb{0,n}(\PP^r,d)$ (see Figure \ref{morb}). 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 $p \in L$.  \medskip

\begin{figure}[htbp]
\begin{center}
\input{morc.pstex_t}
\end{center}
\caption{The morphism $\gamma$.}
\label{morc}
\end{figure}

{\bf The morphism $\gamma$.} If $d \geq 2$, there is a 1-morphism
$\gamma : \PP^1 \rightarrow \Kgnb{0,n}(\PP^r,d)$ 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
$\PP^r$ 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 $L\cong
\PP^1$ gives a stable map parameterized by $\PP^1$, hence there is an
induced 1-morphism $\gamma: \PP^1 \rightarrow \Kgnb{0,n}(\PP^r,d)$
(see Figure \ref{morc}). 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 $L$. \medskip

\begin{notat} \label{notat-Pu}
If $d\geq 2$, denote by $P_{r,n,d}$ the Abelian group
$$
P_{r,n,d} : = \text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \times
\text{Pic}(\PP^{r-1}) \times \text{Pic}(\PP^1)^n \times
\text{Pic}(\PP^1).
$$
Denote by $u=u_{r,n,d}:\text{Pic}(\Kgnb{0,n}(\PP^r,d)) \rightarrow
P_{r,n,d}$ the pull-back map
$$u_{r,n,d} = (\alpha^*,(\beta_1^*,\dots,\beta_n^*),\gamma^*).$$

If $d=1$, denote by $P_{r,n,1}$ the Abelian group
$$ P_{r,n,1} : = \text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \times
\text{Pic}(\PP^{r-1}) \times \text{Pic}(\PP^1)^n
$$ and denote by $u=u_{r,n,1}: \text{Pic}(\Kgnb{0,n}(\PP^r,1))
\rightarrow P_{r,n,1}$ the pull-back map
$$u_{r,n,1} = (\alpha^*,(\beta_1,^*,\dots,\beta_n^*))$$
\end{notat}

Theorem ~\ref{thm-mainpre} is equivalent to the following.

\begin{thm} \label{thm-mainc} The map $u_{r,n,d}\otimes
  \QQ:\text{Pic}(\Kgnb{0,n}(\PP^r,d))_\QQ \rightarrow P_{r,n,d}\otimes
  \QQ$ is an isomorphism.  The image under $u_{r,n,d}\otimes \QQ$ of
  the ample cone, resp. NEF, eventually free cone of
  $\Kgnb{0,n}(\PP^r,d)$ equals the product of the ample cones, resp.
  NEF, eventually free cones of
  $\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$,
  $\text{Pic}(\PP^{r-1})$, and the factors $\text{Pic}(\PP^1)$.
\end{thm}

This is equivalent to Theorem~\ref{thm-mainpre} because the linear map
$u_{r,n,d}$ is simply the inverse of the product of the linear map $v$
and the maps $\QQ \rightarrow \text{Pic}(\Kgnb{0,n}(\PP^r,d))_\QQ$
associated to each generator $\mc{H}$, $\mc{T}$ and
$\mc{L}_1,\dots,\mc{L}_n$.

\begin{notat} \label{notat-triangle} Denote by $\underline{n}$ the set
  $\{1,\dots,n\}$.  Denote by $\triangle = \triangle_{n,d}$ the set of
  $4$-tuples $((A,d_A),(B,d_B))$ of an ordered partition $A\sqcup B$
  of $\underline{n}$ and an ordered pair of nonnegative integers
  $(d_A,d_B)$ such that $d_A+d_B=d$ and $\#A \geq 2$ ($\#B \geq 2$) if
  $d_A = 0$ ($d_B = 0$, respectively).  Denote by $\triangle'$ the
  subset of $\triangle$ of data such that $\#A+d_A\geq 2$ and $\#B+d_B
  \geq 2$.
\end{notat}

Recall that the group $\text{Pic}(\kgnb{0,n})$ is generated by
boundary divisors $\tilde{\Delta}_{A,B}$, where $A \sqcup B$ is an
ordered partition of $\underline{n}$ with $\# A \geq 2$ and $\# B \geq
2$. Let $\tilde{\Delta}_{k,n-k}$ denote the sum of the boundary
divisors $\sum_{(A,B)} \tilde{\Delta}_{A,B}$, where the sum runs over
pairs $(A,B)$ such that $\# A = k$ and $\# B = n-k$. The group
$\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$ is generated by boundary
divisors $\tilde{\Delta}_{(A,d_A),(B,d_B)}$, where $((A,d_A), (B,d_B))
\in \Delta'$. The divisor $\tilde{\Delta}_{(A,d_A),(B,d_B)}$ denotes
the $\mathfrak{S}_d$-invariant sum of boundary divisors $\sum_{(A',
  B')} \tilde{\Delta}_{(A,A'),(B,B')}$, where the sum runs over pairs
$(A',B')$ such that $A' \sqcup B'$ is a partition of the last $d$
points and $\# A' = d_A$ and $\# B' = d_B$.

To apply Theorem~\ref{thm-mainc}, we need to express
the images of the standard generators of
$\text{Pic}(\Kgnb{0,n}(\PP^r,d))$ in terms of the standard generators
for $\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$,
$\text{Pic}(\PP^{r-1})$ and $\text{Pic}(\PP^1)$ factors.

\begin{figure}
$$
\begin{array}{|c|cccc|}
  \hline
  & & & & \\ 
  \mbox{Divisors in} \  \Kgnb{0,n}(\PP^r,d) & \alpha_1^* & \alpha_2^* & \beta_i^* & \gamma^* \\
  \hline
  & & & & \\
  \mc{T}    & 0 & 0 & 0 & \OO_{\PP^1}(2) \\
  & & & & \\
  \mc{H}    & 0 & \OO_{\PP^{r-1}}(d) & 0 & 0 \\
  & & & & \\
  \mc{L}_i  & 0 & 0 & \OO_{\PP^1}(1) & 0 \\
  & & & & \\
  \mc{L}_{j\neq i}  & 0 & 0 & 0 & 0 \\
  & & & & \\
  \Delta_{(\emptyset,1),(\underline{n},d-1)} & c & \OO_{\PP^{r-1}}(-d) &
  \OO_{\PP^1}(-1) & \OO_{\PP^1}(4) \\ 
  & & & & \\
  \Delta_{(\emptyset,2),(\underline{n},d-2)} &
  \tilde{\Delta}_{(\emptyset,2),(\underline{n},d-2)} & 0 & 0 &
  \OO_{\PP^1}(-1) \\
  & & & & \\
  \Delta_{(\{i\},1),(\{i\}^c,d-1)} & \tilde{\Delta}_{(\{i\},1),(\{i\}^c,d-1)} &
  0 & \OO_{\PP^1}(-1) & 0 \\
  & & & & \\
  \Delta_{(A,d_A),(B,d_B)} & \tilde{\Delta}_{(A,d_A),(B,d_B)} & 0 & 0 & 0 \\
  \text{all others} & & & & \\
  \hline
\end{array}
$$
\caption{The pull-backs of the standard generators}
\label{table-1}
\end{figure}

\begin{prop} \label{prop-stdgens}
(i) Assume $d\geq 2$ so that $\gamma$ is defined. Then
  $$\gamma^*\mc{T} =\OO_{\PP^1}(2), \ \ \gamma^*\mc{H}= 0, \ \
  \gamma^*\mc{L}_i = 0, \ \ \mbox{for} \ \ 1\leq i\leq n.$$ The
  pull-back $\gamma^*\Delta_{(A,d_A),(B,d_B)} = 0$ unless $(\#A,d_A)$
  or $(\#B,d_B)$ is equal to $(0,1)$ or $(0,2)$.  Moreover, if
  $(n,d)\neq (0,3)$,
  $$\gamma^*\Delta_{(\emptyset, 1),(\underline{n},d-1)}=
  \OO_{\PP^1}(4) \ \ \mbox{ and} \ \ \gamma^*\Delta_{(\emptyset,
  2),(\underline{n},d-2)} = \OO_{\PP^1}(-1). $$ If $(n,d)=(0,3)$, then
  $\gamma^*\Delta_{(\emptyset, 1),(\underline{n},d-1)}=
  \OO_{\PP^1}(3)$. \smallskip

  (ii) Assume $n\geq 1$ so that $\beta_1,\dots,\beta_n$ are defined.
  Then $$ \beta_i^*\mc{H}= 0, \ \ \beta_i^*\mc{L}_i = \OO_{\PP^1}(1),\
  \ \beta_i^*\mc{L}_j = 0 \ \ \mbox{if} \ \ j\neq i, \ \ \mbox{and} \
  \ \beta_i^*\mc{T} = 0.$$ For every $1\leq i\leq n$, the pull-back
  $\beta_i^*\Delta_{(\emptyset,1),(\underline{n},d-1)}$ equals
  $\OO_{\PP^1}(1)$ if $(n,d)\neq (1,2)$, and equals $\OO_{\PP^1}(2)$
  if $(n,d)=(1,2)$.  If $(n,d) \neq (1,2)$, then
  $\beta_i^*\Delta_{(\{i\},1),(\{i\}^c,d-1)}$ equals
  $\OO_{\PP^1}(-1)$.  And $\beta_i^*\Delta_{(A,d_A),(B,d_B)}$ equals
  $0$ if neither $(A,d_A)$ nor $(B,d_B)$ equal $(\emptyset,1)$ or
  $(\{i\},1)$.


(iii) $\alpha^*\mc{H} = (0,\OO_{\PP^{r-1}}(d)), \ \  \alpha^*\mc{L}_i
  = 0, \ \ \mbox{for} \  1 \leq i \leq n,  \ \  \alpha^*\mc{T} = 0.$

  \noindent If $\#A + d_A, \#B + d_B \geq 2$, then
  $\alpha^*\Delta_{(A,d_A),(B,d_B)}$ equals
  $\tilde{\Delta}_{(A,d_A),(B,d_B)}$.  The pull-back
  $\alpha^*\Delta_{(\emptyset,1),(\underline{n},d-1)}$ equals
  $(c,\OO_{\PP^{r-1}}(-d))$, where $c$ is the class
$$ c = \frac{-1}{(n+d-1)(n+d-2)} \sum_{((A,d_A), (B,d_B)) \atop{ \in
\Delta'}} d_A(d_B+\#B)(d_B+\#B-1) \tilde{\Delta}_{(A,d_A),(B,d_B)}.
$$
\end{prop}

\begin{proof}  
  Items (i) and (ii) follow from Lemma~\ref{lem-alphabeta} and
  Lemma~\ref{lem-TH}.  Item (iii), except for the computation of $c$,
  is straightforward.  The class $c$ equals $-\sum_{i=1}^d
  \psi_{n+i}$.  To rewrite this as above, use ~\cite[Lemma
  2.2.1]{pandh:qdivisors} (cf. also ~\cite[Lemma
  6.10]{jason:virtual}).
\end{proof}


With the exceptions of $(n,d) = (0,3)$, $(1,2)$, and $(1,3)$,
Proposition \ref{prop-stdgens} is
summarized by Figure ~\ref{table-1}.  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 $\gamma^*$ and
$\Delta_{(\emptyset,2),(\underline{n},d-2)}$ apply only if $d\geq 2$.
The lines $\mc{L}_i$, $\mc{L}_j$ and
$\Delta_{(\{i\},1),(\{i\}^c,d-1)}$ only apply if $n\geq 1$.

In ~\cite{kawamata:subadjunction}, Kawamata associated an effective,
NEF $\QQ$-Cartier divisor $\mc{L}$ on $\kgnb{0,n}$ to every $n$-tuple
of rational numbers, $(d_1,\dots,d_n)$ satisfying $0<d_i\leq 1$ and
$d_1+\dots+d_n=2$.  In an unpublished note
Keel and \mac~Kernan proved the following.

\begin{thm}[Keel-\mac~Kernan]  \label{thm-K-McK}
The $\QQ$-Cartier divisor $\mc{L}$ is eventually free.
\end{thm}

In particular, when $d_1=\dots=d_n = 2/n$, the divisor class of
$\mc{L}$ equals $(1/n(n-1))D_n$, where 
$$
D_n = \sum_{k=2}^{\lfloor n/2 \rfloor} k(k-1)
\tilde{\Delta}_{k,n-k}.
$$
This is the divisor class giving the ``democratic'' contraction of the
boundary of $\Kgnb{0,n}$, cf. ~\cite[\S 2.1.2]{Has3}.  
One application of Theorem~\ref{thm-mainc}  is the construction of the
analogous contraction in Theorem~\ref{thm-mainpre2} as well as 
a new, short construction of the democratic 
contraction.

\begin{thm} \label{thm-Mcontract}  
  For every integer $n\geq 4$, there is a contraction
  $$
\text{cont}:\kgnb{0,n} \rightarrow Y
$$ 
restricting to an open
  immersion on the interior $\kgn{0,n}$ and whose restriction to the
  boundary divisor $\Delta_{k,n-k} = \kgn{0,k+1}\times \kgn{0,n+1-k}$
  factors through projection to $\kgn{0,n+1-k}$ for each $3\leq k \leq
  \lfloor n/2 \rfloor$.  The divisor $\mc{D}_n$ is the pull-back of an
  ample divisor on $Y$.
\end{thm}

It follows easily that
for every rational number $b$ satisfying 
$$
\frac{2}{(n-1)}
  < b < \frac{2}{\lfloor n/2 \rfloor} \ \ \ \mbox{ resp.}  \ \ b=
  \frac{2}{\lfloor n/2 \rfloor},
$$ 
setting 
$B=b \
  (\text{cont})_*(\tilde{\Delta}_{2,n-2})$, 
$K_{Y}+B$ is an ample $\QQ$-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 $\mc{L}$).
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section 2
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The splitting homomorphism} \label{sec-split}

\noindent
In this section we define a map
$v:\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \rightarrow
\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$ that maps the NEF divisors
in $\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$ to NEF divisors in
$\Kgnb{0,n}(\PP^r,d)$. The map $v$ gives a splitting of the map
$\alpha_1^*$ defined in the introduction and is essential for the
proof of Theorem~\ref{thm-mainc}. \medskip



Let $\Pi \subset \PP^r$ be a hyperplane not containing the point $p$
used to define the morphisms $\alpha$ and $\gamma$.  Assume that the
degree $d-1$ curve used to define the morphisms $\beta_i$ is not
tangent to $\Pi$, and none of the marked points on this curve are
contained in $\Pi$.  Finally, assume that the degree $d-2$ curve used
to define the morphism $\gamma$ is not tangent to $\Pi$ and none of
the marked points are contained in $\Pi$.
\medskip

Denote by $\Kgn{0,n+d}(\PP^r,d)$ the open substack of
$\Kgnb{0,n+d}(\PP^r,d)$ parameterizing stable maps with irreducible
domain.  Let $$\text{ev}_{n+1,\dots,n+d}:\Kgn{0,n+d}(\PP^r,d)
\rightarrow (\PP^r)^d$$ be the evaluation morphism associated to the last
$d$ marked point. Denote by $\Kgn{0,n+d}(\PP^r,d)_\Pi$ the inverse image
of $\Pi^d$ and  by $\Kgnb{0,n+d}(\PP^r,d)_\Pi$ the
closure of $\Kgn{0,n+d}(\PP^r,d)_\Pi$ in $\Kgnb{0,n+d}(\PP^r,d)$.
\medskip


$\Kgnb{0,n+d}(\PP^r,d)_\Pi$ is $\mathfrak{S}_d$-invariant under the
action of $\mathfrak{S}_d$ on $\Kgnb{0,n+d}(\PP^r,d)$ permuting the
last $d$ marked points.  Denote by $$\pi:\Kgnb{0,n+d}(\PP^r,d)
\rightarrow \Kgnb{0,n}(\PP^r,d)$$ the forgetful 1-morphism that forgets
the last $d$ marked points and stabilizes the resulting family of
prestable maps.  This is $\mathfrak{S}_d$-invariant.  Denote by
$$\rho:\Kgnb{0,n+d}(\PP^r,d) \rightarrow \kgnb{0,n+d}$$ the 1-morphism
that stabilizes the universal family of marked prestable curves over
$\Kgnb{0,n+d}(\PP^r,d)$.  This is $\mathfrak{S}_d$-equivariant.

\begin{lem} \label{lem-pi}  
  The $1$-morphism $\pi:\Kgn{0,n+d}(\PP^r,d)_\Pi \rightarrow
  \Kgnb{0,n}(\PP^r,d)$ is \'etale.  Denoting the image by $O_\Pi$, the
  morphism $\pi:\Kgn{0,n+d}(\PP^r,d)_\Pi \rightarrow O_\Pi$ is an
  $\mathfrak{S}_d$-torsor.
\end{lem}

\begin{proof}  
  Let $(C,(p_1,\dots,p_n,q_1,\dots,q_d),f)$ be a stable map in
  $\Kgn{0,n+d}(\PP^r,d)_\Pi$.  Then $(C,(p_1,\dots,p_n),f)$ satisfies,
\begin{enumerate}
\item[(i)]  
  $C$ is irreducible,
\item[(ii)]  
  $f^{-1}(\Pi)$ is a reduced Cartier divisor, and
\item[(iii)]  
  none of the marked points $p_i$ is contained in $f^{-1}(\Pi)$.
\end{enumerate}
Conversely, for every stable map $(C,(p_1,\dots,p_n),f)$ satisfying
(i)--(iii), for every labeling of $f^{-1}(\Pi)$ as $q_1,\dots,q_d$,
$(C,(p_1,\dots,p_n,q_1,\dots,q_d),f)$ is a stable map in
$\Kgn{0,n+d}(\PP^r,d)_\Pi$.  Thus $O_\Pi$ is the open substack of
stable maps satisfying (i)--(iii) and $\Kgn{0,n+d}(\PP^r,d)_\Pi$ is
the $\mathfrak{S}_d$-torsor over $O_\Pi$ parameterizing labelings of
the fibers of $f^{-1}(\Pi)$.
\end{proof}


Denote by $q:\kgnb{0,n+d} \rightarrow
\kgnb{0,n+d}\slash\mathfrak{S}_d$ the geometric quotient.  The
composition $q\circ \rho:\Kgnb{0,n+d}(\PP^r,d)_\Pi \rightarrow
\kgnb{0,n+d}\slash\mathfrak{S}_d$ is $\mathfrak{S}_d$-equivariant.
Because $\Kgn{0,n+d}(\PP^r,d)_\Pi$ is an $\mathfrak{S}_d$-torsor over
$O_\Pi$, there is a unique 1-morphism $\phi'_{\Pi}:O_\Pi \rightarrow
\kgnb{0,n+d}\slash\mathfrak{S}_d$ such that $\phi'\circ \pi = q\circ
\rho$.

\begin{defn} \label{defn-UV}  
  Define $U_\Pi$ to be the maximal open substack of
  $\Kgnb{0,n}(\PP^r,d)$ over which $\phi'_\Pi$ extends to a
  1-morphism, denoted $$\phi_\Pi:U_\Pi \rightarrow
  \kgnb{0,n+d}\slash\mathfrak{S}_d.$$  Define $I_\Pi$ to be the
  normalization of the closure in $\Kgnb{0,n}(\PP^r,d)\times
  \kgnb{0,n+d}\slash\mathfrak{S}_d$ of the image of the graph of
  $\phi'_\Pi$, i.e., $I_\Pi$ is the normalization of the image of
  $(\pi,q\circ \rho)$.  Define $\widetilde{I}_\Pi$ to be the
  normalization of the image of $(\pi,\rho)$ in $\Kgnb{0,n}(\PP^r,d)
  \times \kgnb{0,n+d}$.  Finally, define $\widetilde{U}_\Pi$ to be the
  inverse image of $U_\Pi$ in $\widetilde{I}_\Pi$.
\end{defn}

\medskip
There is a pull-back map of $\mathfrak{S}_d$-invariant invertible
sheaves, $$\rho^*:\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}
\rightarrow \text{Pic}(\widetilde{I}_\Pi)^{\mathfrak{S}_d},$$ which
further restricts to $\text{Pic}(\widetilde{U}_\Pi)^{\mathfrak{S}_d}$.
After \'etale base-change from $U_\Pi$ to a scheme, the morphism
$\widetilde{U}_\Pi \rightarrow U_\Pi$ is the geometric quotient of
$\widetilde{U}_\Pi$ by the action of $\mathfrak{S}_d$.  Therefore the
pull-back map $\text{Pic}(U_\Pi) \rightarrow
\text{Pic}(\widetilde{U}_\Pi)^{\mathfrak{S}_d}$ is an isomorphism
after tensoring with $\QQ$; in fact, both the kernel and cokernel are
annihilated by $d!$.  Because $\kgnb{0,n+d}\slash\mathfrak{S}_d$ is a
proper scheme and because $\Kgnb{0,n}(\PP^r,d)$ is separated and
normal, by the valuative criterion of properness the complement of
$U_\Pi$ has codimension $\geq 2$. The smoothness of
$\Kgnb{0,n}(\PP^r,d)$ and ~\cite[Prop. 6.5(c)]{hartshorne:book} imply
that the restriction map $\text{Pic}(\Kgnb{0,n}(\PP^r,d)) \rightarrow
\text{Pic}(U_\Pi)$ is an isomorphism.

\begin{defn} \label{defn-v}
Define $v:\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \rightarrow
\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$ to be the unique
homomorphism commuting with $\rho^*$ via the isomorphisms above. 
\end{defn}

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

\begin{lem} \label{lem-NEF}      
  For every base-point-free invertible sheaf $\mc{L}$ in
  $\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$, $v(\mc{L})$ is
  base-point-free.  In particular, for every ample invertible sheaf
  $\mc{L}$, $v(\mc{L})$ is NEF.  Thus, by Kleiman's criterion, for
  every NEF invertible sheaf $\mc{L}$, $v(\mc{L})$ is NEF.
\end{lem}

\begin{proof}  
  For every $[(C,(p_1,\dots,p_n),f)]$ in $\Kgnb{0,n}(\PP^r,d)$, there
  exists a hyperplane $\Pi$ satisfying the conditions above and such
  that $f^{-1}(\Pi)$ is a reduced Cartier divisor containing none of
  $p_1,\dots,p_n$.  By Lemma~\ref{lem-pi}, $(C,(p_1,\dots,p_n),f)$ is
  contained in $U_\Pi$.  Since $\mc{L}$ is base-point-free, there
  exists a divisor $D$ in the linear system $|\mc{L}|$ not containing
  $\phi_\Pi[(C,(p_1,\dots,p_n),f)]$.  By the proof of ~\cite[Prop.
  6.5(c)]{hartshorne:book}, the closure of $\phi_\Pi^{-1}(D)$ is in
  the linear system $|v(\mc{L})|$; and it does not contain
  $[(C,(p_1,\dots,p_n),f)]$.
\end{proof}

\begin{lem} \label{lem-alphabeta}
\begin{enumerate}
\item[(i)]
The images of $\alpha$, $\beta_i$ and $\gamma$ are contained in
$U_\Pi$.
\item[(ii)]  
The morphisms $\phi_{\Pi} \circ \beta_i$ and $\phi_{\Pi} \circ \gamma$
are constant morphisms.  Therefore $\beta_i^* \circ v$ and $\gamma^* \circ
v$ are the zero homomorphism.
\item[(iii)] The composition of $\alpha$ with $\phi_\Pi$ equals $q
\circ \text{pr}_{\kgnb{0,n+d}}$.  Therefore $$\alpha^* \circ
v:\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \rightarrow
\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \times
\text{Pic}(\PP^{r-1})$$ is the homomorphism whose projection on the
first factor is the identity, and whose projection on the second
factor is $0$.
\end{enumerate}
\end{lem}

\begin{proof}  
  \textbf{(i):} The image of $\alpha$ is contained in $O_\Pi$.  Denote
  by $q$ the intersection point of $L$ and $\Pi$.
  
   The image $\beta_i(L-\{q\})$ is contained in
  $O_\Pi$.  The stable map $\beta_i(q)$ sends the $i^\text{th}$ marked
  point into $\Pi$.  Up to labeling the $d$ points of the inverse
  image of $\Pi$, there is only one $(n+d)$-pointed stable map in
  $\Kgnb{0,n+d}(\PP^r,d)_\Pi$ that stabilizes to this stable map.  It
  is obtained from $\beta_i(q)$ by removing the $i^\text{th}$ marked
  point from $L$, attaching a contracted component $C'$ to $L$ at $q$,
  containing the $i^\text{th}$ marked point and exactly one of the
  last $d$ marked points, and labeling the $d-1$ points in $C\cap \Pi$
  with the remaining $d-1$ marked points.
  
       
  Similarly, $\gamma(L-\{q\})$ is contained in $O_\Pi$.  The stable
  map $\gamma(q)$ has two copies of $L$ attached to each other at $q$.
  This appears to be a problem, because the inverse image of
  $\gamma(q)$ in $\Kgnb{0,n+d}(\PP^r,d)_\Pi$ is $1$-dimensional,
  isomorphic to $\kgnb{0,4}$.  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 $C\cap \Pi$.
  However, the map $\rho$ that stabilizes the resulting prestable
  $(n+d)$-marked curve is constant on this $\kgnb{0,4}$.  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 $3$.
  
   \textbf{(ii):} In the family defining $\beta_i$,
  only the $i^\text{th}$ 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 $i^\text{th}$ 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 $\gamma$, 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.
  
     
  \textbf{(iii):} Each stable map in $\alpha(\kgnb{0,n+d}\times
  \PP^{r-1})$ is obtained from a genus $0$, $(n+d)$-pointed, stable
  curve $(C_0,(p_1,\dots,p_n,q_1,\dots,q_d))$ and a line $L$ in
  $\PP^r$ containing $p$ by attaching for each $1\leq i\leq n$, 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 $\PP^r$ 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 $\Pi$, the inverse image of $\Pi$
  consists of the $d$ points $r_1,\dots,r_d$, 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, $(C_0,(p_1,\dots,p_n,q_1,\dots,q_d))$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% Section 4
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{More divisors} \label{sec-2div}

\noindent
In the previous section we constructed a map (see Definition
\ref{defn-v}) $$v:\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}
\rightarrow \text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ.$$ In this
section we prove that the image of $v$,  the divisor
classes $\mc{H}$, $\mc{T}$ and the tautological divisors $\mc{L}_i$,
generate $\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$. \medskip


 The divisor class $\mc{H}_\Lambda$, \cite[Prop. 1]{pandh:qdivisors}
is the class of stable maps whose image intersects a fixed codimension
$2$ linear space $\Lambda$ of $\PP^r$.  This is defined to be the
empty divisor if $r=1$.  For convenience, assume $\Lambda$ is
contained in $\Pi$ and does not intersect $L$ or the curves $C$ used
to define $\beta_i$ and $\gamma$.  If $n\geq 1$, the divisors
$\mc{L}_{i,\Pi}$, $i=1,\dots,n$, \cite[Prop. 1]{pandh:qdivisors} are
the pull-back by $\text{ev}_i$ of the Cartier divisor $\Pi$.  If
$d\geq 1$, the last divisor is $\mc{T}_\Pi$, \cite[\S
2.3]{pandh:qdivisors}; the divisor of stable maps
$(C,(p_1,\dots,p_n),f)$ such that $f^{-1}(\Pi)$ is not a reduced,
finite set of degree $d$.  This is defined to be the empty divisor if
$d=1$.  In ~\cite{pandh:qdivisors} Pandharipande proves that
$\mc{H}_\Lambda$, $\mc{L}_{i,\Pi}$ and $\mc{T}_\Pi$ are irreducible
Cartier divisors (when they are nonempty).

\begin{lem} \label{lem-TH}
\begin{enumerate}
\item[(i)]  
  The Cartier divisors $\mc{T}_\Pi$, $\mc{L}_{i,\Pi}$ and
  $\mc{H}_\Lambda$ are NEF.
\item[(ii)] The pull-backs $\alpha^*(\mc{T}_\Pi)$ and
  $\alpha^*(\mc{L}_{i,\Pi})$ are zero.  The pull-back
  $\alpha^*(\mc{H}_\Lambda)$ equals $(0,\OO_{\PP^{r-1}}(d))$ in
  $\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}\times
  \text{Pic}(\PP^{r-1})$; if $r=1$, then $\OO_{\PP^{r-1}}(1)$ is the
  trivial invertible sheaf.
\item[(iii)] Assume $n\geq 1$ so that $\beta_i$ is defined for $1\leq
  i\leq n$.  The pull-backs $\beta_i^*(\mc{T}_\Pi)$ and
  $\beta_i^*(\mc{H}_\Pi)$ are zero.  For $1\leq j\leq n$ different
  from $i$, $\beta_i^*(\mc{L}_{j, \Pi})$ is zero.  Finally,
  $\beta_i^*(\mc{L}_{i,\Pi})$ is $\OO_{\PP^1}(1)$.
\item[(iv)]  
  Assume $d\geq 2$ so that $\gamma$ is defined.  The pull-backs
  $\gamma^*(\mc{H}_\Lambda)$ and $\gamma^*(\mc{L}_{i,\Pi})$ are zero,
  and $\gamma^*(\mc{T}_\Pi)$ is $\OO_{\PP^1}(2)$ in
  $\text{Pic}(\PP^1)$.
\end{enumerate}
\end{lem}

\begin{proof}    
  \textbf{(i):} By an argument similar to the one in
  Lemma~\ref{lem-NEF}, these divisors are base-point-free (whenever
  they are non-empty). The divisor $\mc{H}_\Lambda$ is big if $r\geq
  2$, and $\mc{T}_\Pi$ is big if $d\geq 2$.  The divisors $\mc{L}_i$
  are not big.
  
     
  \textbf{(ii):} By the proof of Lemma~\ref{lem-alphabeta}, the image
  of $\alpha$ is in $O_\Pi$, which is disjoint from $\mc{T}_\Pi$.
  Also, $\text{ev}_i\circ \alpha$ is the constant morphism with image
  $p$, so the inverse image of $\mc{L}_i$ is empty.  Finally, the
  pull-back of $\mc{H}_\Pi$ equals the pull-back under the diagonal
  $\Delta$ of the Cartier divisor $\sum_{j=1}^d
  \text{pr}_j^{-1}(\Lambda)$ in $(\PP^{r-1})$, where $\Lambda$ is
  considered as a divisor in $\PP^{r-1}$ via projection from $p$.
  
\medskip  \textbf{(iii):} Since the image of $\beta_i$ is
disjoint from  $\mc{H}_{\Pi}$, $\mc{T}_{\Pi}$ and $\mc{L}_{j,\Pi}$ for
$j \not= i$, the corresponding pull-backs are zero. The map $ev_i
\circ \beta_i : \PP^1 \rightarrow \PP^r$ embeds $\PP^1$ as the line
$L$ in $\PP^r$, hence $\beta_i^* (\mc{L}_{i,\Pi}) = \OO_{\PP^1}(1)$.

    
  \textbf{(iv):} Since neither the image curve nor the marked points
  vary under $\gamma$, clearly $\gamma^*\mc{H}_{\Lambda}$ and
  $\gamma^*\mc{L}_{i,\Pi}$ are zero.  To compute $\gamma^*\mc{T}_\Pi$,
  use ~\cite[Lem 2.3.1]{pandh:qdivisors}.
\end{proof}


The main observation  of this section is the following.

\begin{prop} \label{prop-surj}
  The $\QQ$-vector space $\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$
  is generated by $\mc{T}_\Pi$, $\mc{H}_\Lambda$, $\mc{L}_{i,\Pi}$ for
  $1\leq i\leq n$, and the image of $v$.
\end{prop}

\begin{proof} When $r \geq 2$, Pandharipande  proves that the classes
  of the divisors $\mc{H}_\Lambda$, $\mc{L}_{i,\Pi}$ for $1\leq i\leq
  n$, and the boundary divisors $\Delta_{(A,d_A),(B,d_B)}$ for
  $((A,d_A),(B,d_B)) \in \Delta$ generate the $\QQ$-vector space
  $\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$,
  cf. \cite[Prop. 1]{pandh:qdivisors}. The tangency divisor $\mc{T}$
  can be expressed in terms of $\mc{H}$ and the boundary divisors as
  follows \cite[Lem 2.3.1]{pandh:qdivisors}: 
$$\mc{T} = \frac{d-1}{d}\mc{H} + \sum_{j=0}^{\lfloor \frac{d}{2}
    \rfloor} \frac{j(d-j)}{d} \sum_{((A,d_A),(B,d_B)), d_A =j}
    \Delta_{ (A,d_A),(B,d_B)}.$$ 


\noindent From Lemmas \ref{lem-TH} and \ref{lem-alphabeta} and by
    pairing with one-parameter families, we see that
    $$v(\tilde{\Delta}_{(A,d_A), (B,d_B)}) = \Delta_{(A,d_A),
    (B,d_B)}$$ unless $(\#A, d_A)$ or $(\# B, d_B)$ equals one of
    $(0,2)$ or $(1,1)$.
    $$v(\tilde{\Delta}_{(A,d_A), (B,d_B)}) = \frac{1}{2} \ \mc{T} +
    \Delta_{(A,d_A), (B,d_B)}$$ if $(\#A,d_A)$ or $(\# B, d_B)$ equals
    $(0,2)$. Finally, $$v(\tilde{\Delta}_{(\{ i \},1), (\{ i \}^c,
    d-1)} ) = \Delta_{(\{ i \},1), (\{ i \}^c, d-1)} + \mc{L}_{i,
    \Pi}.$$ Consequently, it follows that the classes of the divisors
    $\mc{H}, \mc{T}$, $\mc{L}_{i,\Pi}$ and the image of $v$ generate
    the classes of all the boundary divisors in the Kontsevich moduli
    space. Hence, they generate
    $\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$. \medskip

We can reduce the case $r=1$ to the case $r \geq 2$. Because $L$ is
disjoint from $\Lambda$, there is a unique linear projection
$$\text{pr}_{\Lambda}:(\PP^r - \Lambda) \rightarrow L$$ whose
restriction to $L$ is the identity.  This is a vector bundle over $L$
whose associated sheaf of sections is $\OO_L(1)^{\oplus (r-1)}$.
Composing a stable map to $(\PP^r-\Lambda)$ with $\text{pr}_\Lambda$
gives a stable map to $L$.  This defines a 1-morphism,
$$
\Kgnb{0,n}(\text{pr}_\Lambda,d):(\Kgnb{0,n}(\PP^r,d) -
\mc{H}_\Lambda) \rightarrow \Kgnb{0,n}(L,d).
$$
This is a vector bundle over $\Kgnb{0,n}(L,d)$ whose associated sheaf
of sections is the sheaf whose fiber at $(C,(p_1,\dots,p_n),f)$ equals
$H^0(C,f^*\OO_L(1)^{\oplus (r-1)})$.  Thus the pull-back homomorphism,
$$ \Kgnb{0,n}(\text{pr}_\Lambda,d)^*:\text{Pic}(\Kgnb{0,n}(L,d))
\rightarrow \text{Pic}(\Kgnb{0,n}(\PP^r,d) - \mc{H}_\Lambda),
$$
is an isomorphism, cf. ~\cite[Thm. 3.3(a)]{fulton:intersection}.

 The hyperplane $\Pi$ is the closure of
$\text{pr}_\Lambda^{-1}(L\cap \Pi)$.  Thus $U_\Pi - \mc{H}_\Lambda\cap
U_\Pi$ (see Definition \ref{defn-UV}) is the inverse image of the
corresponding open substack of $\Kgnb{0,n}(L,d)$ for $L\cap \Pi$
inside $L$.  The inverse image of $\mc{T}_{L\cap \Pi}$, resp.
$\mc{L}_{i,L\cap \Pi}$, equals the restriction of $\mc{T}_\Pi$, resp.
$\mc{L}_{i,\Pi}$.  And $\phi_{L\cap \Pi} \circ
\Kgnb{0,n}(\text{pr}_\Lambda,d)$ equals the restriction of $\phi_\Pi$.
Thus $\text{Pic}(\Kgnb{0,n}(\PP^r,d) - \mc{H}_\Lambda)\otimes \QQ$ is
generated by $\mc{T}_\Pi$, $\mc{L}_{i,\Pi}$ for $1\leq i\leq n$, and
the image of $v$ if and only if the same is true for
$\text{Pic}(\Kgnb{0,n}(\PP^1,d))\otimes \QQ$.

\end{proof}





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%% Section 5
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\section{Proof of the main theorem} \label{sec-proof}

\noindent
In this section we complete the proof of
Theorem~\ref{thm-mainc}. Recall that Theorem~\ref{thm-mainc} asserts
that the NEF cone of the Kontsevich moduli space $\Kgnb{0,n}(\PP^r,d)$
equals the NEF cone in $P_{r,n,d} \otimes \QQ$, where $P_{r,n,d}$ is
the Abelian group $$ P_{r,n,d} : =
\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d} \times \text{Pic}(\PP^{r-1})
\times \text{Pic}(\PP^1)^n \times \text{Pic}(\PP^1).
$$ The identification of $\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$
with $P_{r,n,d}\otimes \QQ$ is given by the map
$$u = u_{r,n,d} : = (\alpha^*,(\beta_1^*,\dots,\beta_n^*),\gamma^*)$$
(see \S \ref{sec-intro}). \medskip

 Denote by $$\widetilde{v}:P_{r,n,d}\otimes \QQ \rightarrow
\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$$ the unique homomorphism
whose restriction to $\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$ is
$v$ (see Definition \ref{defn-v}) , whose restriction to
$\text{Pic}(\PP^{r-1})$ sends $\OO_{\PP^{r-1}}(1)$ to
$[\mc{H}_\Lambda]$, whose restriction to the $i^\text{th}$ factor of
$\text{Pic}(\PP^1)^n$ sends $\OO_{\PP^1}(1)$ to $[\mc{L}_i]$ if $n\geq
1$,  and whose restriction to
the last factor $\text{Pic}(\PP^1)$ (assuming $d\geq 2$) sends
$\OO_{\PP^1}(1)$ to $1/2 \ [\mc{T}_\Pi]$.  By
Lemma~\ref{lem-alphabeta} (ii), (iii) and by Lemma~\ref{lem-TH},
$u\otimes \QQ \circ \widetilde{v}$ is the identity map.  In
particular, $\widetilde{v}$ is injective.  By
Proposition~\ref{prop-surj}, $\widetilde{v}$ is surjective.  Thus
$\widetilde{v}$ and $u\otimes \QQ$ are isomorphisms.

 
Because $\alpha$, $\beta_i$ and $\gamma$ are morphisms, for every NEF,
resp. eventually free, divisor $D$ in
$\text{Pic}(\Kgnb{0,n}(\PP^r,d))\otimes \QQ$, $\alpha^*(D)$,
$\beta_i^*(D)$, and $\gamma^*(D)$ are NEF, resp. eventually free.
Denote,
$$
D_1 = \alpha_1^*(D), \ \
a\ [\OO_{\PP^{r-1}}(1)] = \alpha_2^*(D), \ \ 
b_i\ [\OO_{\PP^1}(1)] = \beta_i^*(D), \ \ 
c\ [\OO_{\PP^1}(1)] = \gamma^*(D),
$$
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
$\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$, and $a,b_i,c\geq 0$.


Conversely, by Lemma~\ref{lem-NEF}, for every NEF, resp. eventually
free, divisor $D_1$ in $\text{Pic}(\kgnb{0,n+d})^{\mathfrak{S}_d}$,
$v(D_1)$ is NEF, resp. eventually free.  By Lemma~\ref{lem-TH}(i), for
$a,b_i,c\geq 0$, $a[\mc{H}_\Lambda]$, $b_i[\mc{L}_{i,\Pi}]$ and $c/2\
[\mc{T}_\Pi]$ are NEF and eventually free.  Since a sum of NEF,
resp. eventually free, divisors is NEF, resp. eventually free, $D =
v(D_1) + a \ [\mc{H}_\Lambda] + b_i \ [\mc{L}_i] + c/2 \ [\mc{T}_\Pi]$
is NEF, resp. eventually free.  Therefore $D$ is NEF if and only if
$u\otimes \QQ(D)$ 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
$u\otimes \QQ(D)$ is contained in the product of the ample cones of
the factors. $\Box$

\medskip

\begin{rmk} Since the analogue of the F-conjecture is
known for $\kgnb{0,d}\slash\mathfrak{S}_d$ when $d \leq 11$ by
~\cite{Keel-McK}, Theorem \ref{thm-mainc} provides an explicit
description of the NEF cone of $\Kgnb{0,0}(\PP^r,d)$ for $r \geq 2$
and $d \leq 11$. For example, when $d=2,3$, the NEF cone is bounded by the rays
$\mc{H}$ and $\mc{T}$.  When $d=4,5$,
the NEF cone is generated by the rays $\mc{H}, \  \mc{T}$ and
$\mc{H} + \Delta_{1, d-1} + 4 \ \Delta_{2,d-2}.$ \end{rmk}




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%% Section 6
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The contractions} \label{sec-contract}

Theorem~\ref{thm-Mcontract}(i)--(v) and Theorem~\ref{thm-mainpre2}
are proved simultaneously by induction on $n$, resp. $d$, in the
following 2 lemmas.  Since the divisor $D_n =\sum_{k=2}^{\lfloor n/2
\rfloor} k(k-1)\tilde{\Delta}_{k,n-k}$ is ample on $\kgnb{0,n}$ for $n=4,5$,
the base cases $n=4,5$ for Theorem~\ref{thm-Mcontract} are immediate.
The base cases $d=2,3$ for Theorem~\ref{thm-mainpre2} are also
straightforward.

\begin{lem} \label{lem-cont1}
Let $d\geq 4$ be an integer.
The existence of a contraction as in
Theorem~\ref{thm-Mcontract} for $n=d$ implies
the existence of a contraction as in Theorem~\ref{thm-mainpre2} for $d$.
\end{lem}  

\begin{proof}
The divisor $D_{r,d}= \mc{T} + \sum_{k=2}^{\lfloor d/2 \rfloor}
  k(k-1)\Delta_{k,d-k}$ equals $v(D_d)$.  By hypothesis, $v(D_d)$ is
eventually free, thus $D_{r,d}$ is eventually free by
Lemma~\ref{lem-NEF}.  Define $\text{cont}_{r,d}:\Kgnb{0,0}(\PP^r,d)
\rightarrow Y_{r,d}$ 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 $\text{cont}_{r,d}$ is finite.  The
claim is that $\text{cont}_{r,d}^{-1}(O_{r,d})$ contains
$\Kgn{0,0}(\PP^r,d)$.  Every proper, irreducible curve $B$ in
$\Kgnb{0,0}(\PP^r,d)$ contracted by $\text{cont}_{r,d}$ has
intersection number $0$ with $D_{r,d}$.  If $B$ intersects
$\Kgn{0,0}(\PP^r,d)$, the intersection number with every
$\Delta_{k,d-k}$, $k=1,\dots,\lfloor d/2 \rfloor$ is nonnegative.
Since $\mc{T}$ is NEF, the intersection number of $B$ with $\mc{T}$ is
nonnegative.  Since $D_{r,d} \cdot B =0$, $B$ has intersection number
zero with $\mc{T}$ and $\Delta_{k,d-k}$, $k=2,\dots,\lfloor d/2
\rfloor$. From the expression of $\mc{T}$ and the fact that $\mc{H}$
and $\Delta_{1,d-1}$ have nonnegative intersection with $B$, it
follows that $B$ has intersection number zero with $\mc{H}$ and
$\Delta_{1,d-1}$, as well.  Since there exists an ample linear
combination of these divisors, we obtain a contradiction. Thus $B$ is
contained in the complement of $\Kgn{0,0}(\PP^r,d)$, proving the
claim.


By Zariski's Main Theorem,
$\text{cont}_{r,d}:\text{cont}_{r,d}^{-1}(O_{r,d})\rightarrow O_{r,d}$
is an isomorphism.  In particular,
$\text{cont}_{r,d}:\Kgn{0,0}(\PP^r,d) \rightarrow O_{r,d}$ is an open
immersion.  

The 1-morphism $\phi_{\Pi}$ from
Definition~\ref{defn-UV} maps $\Delta_{k,d-k}$ to $\tilde{\Delta}_{k,d-k}$
compatible with the boundary maps.  Thus $\text{cont}_{r,d}$ satisfies
the conclusion of Theorem~\ref{thm-mainpre2}.
\end{proof}  

\begin{lem} \label{lem-cont2}
Let $n\geq 6$ be an integer.  The existence of a contraction as in
Theorem~\ref{thm-mainpre2} for $d=n-2$
implies the existence of a contraction as in
Theorem~\ref{thm-Mcontract} for $n$.  
\end{lem}

\begin{proof}
Denote by $\text{ev}:\Kgnb{0,n}(\PP^{n-2},n-2) \rightarrow
(\PP^{n-2})^{n}$ the evaluation 1-morphism.  Denote by
$\Phi:\Kgnb{0,n}(\PP^{n-2},n-2) \rightarrow \kgnb{0,n}$ the forgetful
1-morphism.  Denote by $U\subset (\PP^{n-2})^{n}$ the open subset
parameterizing $n$-tuples of points in linear general position,
i.e., the span of every $(n-1)$-tuple equals $\PP^{n-2}$.  Kapranov
proves that the 1-morphism,
$$
(\text{ev},\Phi):\Kgnb{0,n}(\PP^{n-2},n-2) \rightarrow (\PP^{n-2})^n
\times \kgnb{0,n},
$$
is an isomorphism over $U \times \kgnb{0,n}$,
~\cite{kapranov:veronese}.  Fix a general point $q$ in $U$, and
identify $\kgnb{0,n}$ with the fiber over $\{q\} \times \kgnb{0,n}$.

The forgetful 1-morphism $\pi:\Kgnb{0,n}(\PP^{n-2},n-2) \rightarrow
\Kgnb{0,0}(\PP^{n-2},n-2)$ restricts to a 1-morphism $p:\kgnb{0,n}
\rightarrow \Kgnb{0,0}(\PP^{n-2},n-2)$.  Denote by
$\text{cont}:\kgnb{0,n} \rightarrow Y$ the Stein factorization of,
$$
\text{cont}_{n-2,n-2}\circ p:\kgnb{0,n} \rightarrow
\text{cont}_{n-2,n-2}(p(\kgnb{0,n})).
$$ 
It is straightforward that $p^{-1}(\Delta_{k-1,n-1-k}) =
\tilde{\Delta}_{k,n-k}$ for every $2\leq k \leq \lfloor n/2 \rfloor$
compatibly with the boundary maps.  Thus $\text{cont}$ satisfies
the conclusion of Theorem~\ref{thm-Mcontract}.  
\end{proof}

\begin{rmk} \label{rmk-cont}
Pairing with test curves gives that 
$$
v\left( \frac{1}{n-1} D_n
\right) = \frac{1}{n-1}D_{r,n} \ \ \mbox{and} \ \ p^* \left(
\frac{1}{n-3}D_{n-2,n-2} \right) = \frac{1}{n-1}D_n.
$$ 
The image of the ample cone of $\Kgnb{0,0}(\PP^n,n)$
under $p^*$ is not all of the ample cone of $\kgnb{0,n+2}$, already
for $n=6$. 
\end{rmk}

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