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\title{The effective 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.}

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\begin{abstract}
In this paper we prove that the cone of effective divisors on the
Kontsevich moduli spaces of stable maps $\Kgnb{0,0}(\PP^r,d)$
stabilize when $r \geq d$. We give a complete characterization of the
effective divisors on $\Kgnb{0,0}(\PP^d,d)$: They are non-negative
linear combinations of boundary divisors and the divisor of maps with
degenerate image.
\end{abstract}



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


The ample and effective cones of divisors play a crucial role in the
birational geometry of a variety. The study of these cones for the
moduli spaces of stable curves has been especially fruitful leading to
the proof that the moduli space of stable curves $\overline{M}_g$ is
of general type when $g > 23$ (see \cite{joe:mumford:generaltype},
\cite{joe:generaltype2}, \cite{joe:eisenbud:generaltype}). Recently,
inspired by the work of G. Farkas, D. Khosla and M. Popa, there has
been renewed interest in constructing divisors of small slope on
$\overline{M}_g$ in order to understand the effective cone of
$\overline{M}_g$ and to determine the Kodaira dimension of
$\overline{M}_g$ in the remaining cases (see \cite{popa:k3},
\cite{gabi:syzygies}). \medskip

The aim of this paper is to describe the classes of effective divisors
on a related moduli space, the Kontsevich moduli space of stable maps
$\Kgnb{0,0}(\PP^r,d)$. For $d>1$, the scheme parameterizing smooth,
degree $d$, rational curves in $\PP^r$ is not proper. The Kontsevich
moduli space gives a useful compactification.  For integers $n,d \geq
0$, the Kontsevich moduli space $\Kgnb{0,n}(\PP^r,d)$ is a smooth,
proper, Deligne-Mumford stack parameterizing the data $(C,(p_1,...,p_n),f)$
of,
\begin{enumerate}
\item[(i)]
  $C$, a proper, connected, at-worst-nodal curve of arithmetic genus
  $0$,
\item[(ii)]  
  $p_1,\dots,p_n$, an ordered sequence of distinct, smooth points of
  $C$,
\item[(iii)]  
  and $f:C\rightarrow \PP^r$, a morphism with $\text{deg}(f^*
  \OO_{\PP^r}(1)) = d$ 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 $C$.
\end{enumerate}
In this paper we will determine the classes of all effective divisor
on $\Kgnb{0,0}(\PP^r,d)$ when $r \geq d$.

\medskip 

In ~\cite{pandh:qdivisors} R. Pandharipande proves that when $r \geq
2$, the divisor class $\mc{H}$, and the classes of the boundary divisors
$\Delta_{k,d-k}$ for $1 \leq k \leq \lfloor d/2 \rfloor$ generate the
group of $\QQ$-Cartier divisors of $\Kgnb{0,0}(\PP^r,d)$. We recall
that

\begin{enumerate}
\item $\mc{H}$ is the class of the divisor of maps whose images intersect
a fixed codimension two linear space in $\PP^r$ (provided $r>1$ and $d
> 0$).


\item $\Delta_{k, d-k}$, $1 \leq k \leq \lfloor d/2 \rfloor$, is the
  class of the boundary divisor consisting of maps with reducible
  domains, where the map has degree $k$ on one component and degree
  $d-k$ on the other component.
\end{enumerate}
The main problem we would like to address is the following:

\begin{problem} \label{prob-main}
Describe the cone of effective divisor classes on
$\Kgnb{0,0}(\PP^r,d)$ in terms of these generators of the Picard
group.
\end{problem}

\medskip\noindent Denote by $P_d$ the $\QQ$-vector space of dimension
$\lfloor d/2 \rfloor + 1$ with basis labeled $\mc{H}$ and
$\Delta_{k,d-k}$ for $k=1,\dots,\lfloor d/2 \rfloor$.  For each $r\geq
2$, there is a $\QQ$-linear map $$u_{d,r}:P_d \rightarrow
\text{Pic}(\Kgnb{0,0}(\PP^r,d))\otimes \QQ$$ that is an isomorphism of
$\QQ$-vector spaces.

\begin{definition} \label{defn-eff}
For every integer $r\geq 2$, denote by $\text{Eff}_{d,r} \subset P_d$
the inverse image under $u_{d,r}$ of the effective cone of
$\Kgnb{0,0}(\PP^r,d)$.  
\end{definition}

\medskip\noindent A more precise version of Problem~\ref{prob-main} is
to describe $\text{Eff}_{d,r}$. A first result is that for a fixed
degree $d$, there is an inclusion between these cones as $r$
increases. Furthermore, the cones stabilize for $r \geq d$.

\begin{proposition} \label{prop-stab}
For every integer $r\geq 2$, $\text{Eff}_{d,r}$ is contained in
$\text{Eff}_{d,r+1}$.  For every integer $r\geq d$, $\text{Eff}_{d,r}$
equals $\text{Eff}_{d,d}$.
\end{proposition}

In view of Proposition \ref{prop-stab} it is especially interesting to
understand $\text{Eff}_{d,d}$.  Most of our paper will concentrate on
this case. \medskip

The crudest invariant one can associate to the effective cone is the
slope of distinguished rays.  For example, Harris and Morrison in
\cite{joe:slope} define the slope of $\overline{M}_g$ as the slope of
the ray that bounds the effective cone in the subspace spanned by the
Hodge class $\lambda$ and the total boundary class
$\delta$. Determining the slope of $\overline{M}_g$ is a major open
problem. In analogy with the case of $\overline{M}_g$, we define the
slope $s(r,d)$ of the effective cone of $\Kgnb{0,0}(\PP^r,d)$ as
follows.


$$s(r,d) := \sup_{\alpha}\ \{ \ \alpha \ : \ \mc{H} - \alpha
\sum_{k=1}^{\lfloor d/2 \rfloor} k (d-k) \ \Delta_{k,d-k} \ \ \text{is
on the effective cone} \ \}.$$ It is possible to determine the slope
for the Kontsevich moduli spaces in the stable range.

\begin{theorem}\label{slope}
If $r \geq d$, then the slope $s(r,d)$ of the effective cone of
$\Kgnb{0,0}(\PP^r,d)$ is equal to $$ s(r,d) = \frac{1} {d+1}.$$
\end{theorem}

When $r=d$, the effective divisor that achieves the extremal slope has
a simple description. Let $D_{\text{deg}}$ denote the locus
parameterizing stable maps $f: C \rightarrow \PP^d$ of degree $d$
whose set theoretic image does not span $\PP^d$. $D_{\text{deg}}$ is a
divisor in $\Kgnb{0,0}(\PP^d,d)$ and has the desired slope. \medskip


$D_{\text{deg}}$ plays a crucial role in describing the effective cone
of $\Kgnb{0,0}(\PP^d,d)$. The following theorem, which describes the
effective cone of $\Kgnb{0,0}(\PP^d,d)$ completely, is the main
theorem of our paper.


\begin{theorem}\label{mainthm}
The class of a divisor lies in the effective cone of
$\Kgnb{0,0}(\PP^d,d)$ if and only if it is a non-negative linear
combination of the class of $D_{\text{deg}}$ and the classes of the
boundary divisors $\Delta_{k,d-k}$ for $1 \leq k \leq \lfloor d/2
\rfloor$.
\end{theorem}

\noindent Theorem \ref{slope} follows immediately from Theorem
\ref{mainthm}. However, since it is easy to give an independent proof
and since the curves that span the null-space of the divisor
$D_{\text{deg}}$ are interesting in their own right, we will give a
simple proof of it in \S \ref{prelim}. Combining Theorem \ref{mainthm}
with Proposition \ref{prop-stab} and Lemma \ref{theclass} we obtain
the following corollary.


\begin{corollary}
When $r \geq d$, the class of a divisor lies in the effective cone of
$\Kgnb{0,0}(\PP^r,d)$ if and only if it is a non-negative linear
combination of the class $$ \mc{H} - \frac{1}{d+1} \sum_{k=1}^{\lfloor
d/2 \rfloor} k (d-k) \Delta_{k,d-k}$$ and the classes of the boundary
divisors $\Delta_{k,d-k}$ for $1 \leq k \leq \lfloor d/2 \rfloor$.
\end{corollary}

The space of curves of a given degree and genus has many distinguished
subvarieties defined by imposing geometric conditions on the
curves. Examples of such subvarieties are given by curves that have an
unexpected secant linear space or curves with an unexpected osculating
linear space or curves with a point of unexpected ramification
sequence. An informal way of restating Theorem \ref{mainthm} is to say
that ``geometric conditions'' do not give new divisors on the space of
rational curves of degree $d$ in $\PP^d$. Rational normal curves are
too predictable. \medskip

We now briefly outline the proof of Theorem \ref{mainthm}. Since
$D_{\text{deg}}$ and the boundary divisors are effective, any
non-negative rational linear combination of these divisors lies in the
effective cone. The main content of the theorem is to show that there
are no other effective divisor classes.

\begin{definition}
A reduced, irreducible curve $C$ on a scheme $X$ is a {\it moving
curve} if the deformations of $C$ cover a Zariski open subset of
$X$. More precisely, a curve $C$ is a moving curve if there exists a
flat family of curves $\pi: \mathcal{C} \rightarrow T$ on $X$ such
that $\pi^{-1}(t_0) = C$ for $t_0 \in T$ and for a Zariski open subset
$U \subset X$ every point $x \in U$ is contained in $\pi^{-1}(t)$ for
some $t \in T$. We call the class of a moving curve a {\it moving
curve class}.
\end{definition}
An obvious observation is that the intersection pairing between the
class of an effective divisor and a moving curve class is always
positive. Intersecting divisors with a moving curve class gives an
inequality for the coefficients of an effective divisor class. The
strategy for the proof of Theorem \ref{mainthm} is to produce enough
moving curves to force the effective divisor classes to be a
non-negative linear combination of $D_{\text{deg}}$ and the boundary
classes.
\medskip

\noindent 
Moving curves in $\Kgnb{0,0}(\PP^d,d)$ are easy to recognize by the
following lemma.

\begin{lemma}\label{moving}
If $C \subset \Kgnb{0,0}(\PP^d,d)$ is a reduced, irreducible curve
that intersects the complement in $\Kgnb{0,0}(\PP^d,d)$ of the
boundary divisors and the divisor of maps whose image is degenerate,
then $C$ is a moving curve.
\end{lemma}

\begin{proof} The automorphism group of $\PP^d$ acts transitively on
  rational normal curves. An irreducible curve of degree $d$ that
  spans $\PP^d$ is a rational normal curve. Hence, a curve $C \subset
  \Kgnb{0,0}(\PP^d,d)$ that intersects the complement in
  $\Kgnb{0,0}(\PP^d,d)$ of the boundary divisors and the divisor of
  maps whose image is degenerate, contains a point that represents a
  map that is an embedding of $\PP^1$ as a rational normal curve. The
  translations of $C$ by $\PP GL (d+1)$ cover a Zariski open set of
  $\Kgnb{0,0}(\PP^d,d)$.
\end{proof}

In \S \ref{proof} using certain linear systems on blow-ups of $\PP^1
\times \PP^1$ we will construct one-parameter families of rational
curves whose general member is a rational normal curve. By Lemma
\ref{moving} these will be moving curves in
$\Kgnb{0,0}(\PP^d,d)$. These moving curves will give us enough
inequalities on the effective cone to deduce Theorem \ref{mainthm}.
\smallskip

Finally, we remark that most of the discussion in this paper extends
to spaces of stable maps to homogeneous varieties such as the
Grassmannians, and flag varieties. In a forthcoming paper we will
explain these generalizations \cite{coskun:homogeneous}.
\bigskip

\noindent {\bf Acknowledgments:} We would like to thank A. J. de Jong,
B. Hassett and R. Miranda for useful discussions.

\section{Preliminaries}\label{prelim}

In this section we prove Proposition \ref{prop-stab} and collect the
basic facts that we will need about the divisor $D_{\text{deg}}$.

\subsection{The stability of the effective cone}

In this subsection we prove that $\text{Eff}_{d,r}$ is contained in
$\text{Eff}_{d, r+1}$ and that $\text{Eff}_{d,r} = \text{Eff}_{d,d}$
for $r \geq d$. Recall that $\text{Eff}_{d,r}$ is the image of the
effective cone of $\Kgnb{0,0}(\PP^r,d)$ when the rational Picard group
of $\Kgnb{0,0}(\PP^r,d)$ is identified with the vector space that has
a basis labeled by $\mc{H}$ and $\Delta_{k,d-k}$ for $1 \leq k \leq
\lfloor d/2 \rfloor$.


\begin{proof}[Proof of Proposition~\ref{prop-stab}]
Let $p\in \PP^{r+1}$ be a point, denote $U = \PP^{r+1} - \{p\}$, and
let $\pi:U \rightarrow \PP^r$ be a linear projection from $p$.  This
induces a smooth 1-morphism $$\Kgnb{0,0}(\pi,d):\Kgnb{0,0}(U,d)
\rightarrow 
\Kgnb{0,0}(\PP^r,d).$$  Let $i:U\rightarrow \PP^{r+1}$ be the open
immersion.  This induces a 1-morphism $$\Kgnb{0,0}(i,d):\Kgnb{0,0}(U,d)
\rightarrow \Kgnb{0,0}(\PP^{r+1},d)$$ relatively representable by open
immersions.  The complement of the image of 
$\Kgnb{0,0}(i,d)$ has codimension $r$, which is greater than $2$.
Therefore, the pull-back morphism
$$\Kgnb{0,0}(i,d)^*:\text{Pic}(\Kgnb{0,0}(\PP^{r+1},d)) \rightarrow
\text{Pic}(\Kgnb{0,0}(U,d))$$ is an isomorphism.  So there is a unique
homomorphism $$h:\text{Pic}(\Kgnb{0,0}(\PP^r,d)) \rightarrow
\text{Pic}(\Kgnb{0,0}(\PP^{r+1},d))$$ such that $$\Kgnb{0,0}(\pi,d)^* =
\Kgnb{0,0}(i,d)^*\circ h.$$  

Recalling from the introduction that $u(r,d)$ is the map that
identifies the Picard group of $\Kgnb{0,0}(\PP^r,d))$ with the vector
space spanned by $\mc{H}$ and the boundary divisors $\Delta_{k,d-k}$,
we see that  $h\circ
u_{d,r}$ equals $u_{d,r+1}$.  So to prove $\text{Eff}_{d,r}$ is
contained in $\text{Eff}_{d,r+1}$, it suffices to prove that
$\Kgnb{0,0}(\pi,d)$ pulls back effective divisors to effective
divisors classes, which follows since $\Kgnb{0,0}(\pi,d)$ is smooth.

\medskip\noindent
Next assume $r\geq d$.  Let $D$ be any effective divisor in
$\Kgnb{0,0}(\PP^r,d)$.  A general point in the complement of $D$
parameterizes a stable map $f:C\rightarrow \PP^r$ such that $f(C)$
spans a $d$-plane.  
Denote by $j:\PP^d \rightarrow \PP^r$ a linear embedding whose
image is this $d$-plane.  There is an induced 1-morphism
$$\Kgnb{0,0}(j,d):\Kgnb{0,0}(\PP^d,d) \rightarrow
\Kgnb{0,0}(\PP^r,d).$$ The map $\Kgnb{0,0}(j,d)^*\circ u_{d,r}$ equals
$u_{d,d}$.  By construction, $\Kgnb{0,0}(j,d)^*([D])$ is the class of
the effective divisor $\Kgnb{0,0}(j,d)^{-1}(D)$, i.e., $[D]$ is in
$\text{Eff}_{d,d}$.  Thus $\text{Eff}_{d,d}$ contains
$\text{Eff}_{d,r}$, which in turn contains $\text{Eff}_{d,d}$ by the
last paragraph.  Therefore $\text{Eff}_{d,r}$ equals
$\text{Eff}_{d,d}$.
\end{proof}


\subsection{The divisor class $D_{\hbox{deg}}$}

In this subsection we determine the class of the divisor of degenerate
maps in $\Kgnb{0,0}(\PP^r,d)$. We then give a basis of moving curves
that span the null-space of $D_{\text{deg}}$ in the cone of
curves. This completes the proof of Theorem \ref{slope}.


\begin{lemma}\label{theclass}
The class $D_{\text{deg}}$ equals
\begin{equation}\label{cldeg}
D_{\text{deg}} = \frac{1}{2d}\lt[ (d+1)\mc{H} - \sum_{k=1}^{\lfloor
d/2 \rfloor} k(d-k) \Delta_{k,d-k}\rt].
\end{equation}
\end{lemma}

\begin{proof} 
We will prove the equality (\ref{cldeg}) by intersecting $D_{\deg}$ by
test curves. Fix a general rational normal scroll of degree $i$ and a
general rational normal curve of degree $d-i-1$ intersecting the
scroll in one point $p$. Consider the one-parameter family $C_i$ of
degree $d$ curves consisting of the fixed degree $d-i-1$ rational
normal curve union curves in a general pencil (that has $p$ as a
base-point) of degree $i+1$ rational normal curves on the scroll. When
$ 2 \leq i \leq \lfloor d/2 \rfloor$, $C_i$ has the following
intersection numbers with $\mc{H}$ and $D_{\text{deg}}$.  $$C_i \cdot
\mc{H} = i, \ \ C_i \cdot D_{\text{deg}} = 0.$$ The curve $C_i$ is
contained in the boundary divisor $\Delta_{i+1, d-i-1}$ and has
intersection number $$C_i \cdot \Delta_{i+1, d-i-1} = -1$$ with
it. The intersection number of $C_i$ with the boundary divisors
$\Delta_{i,d-i}$ and $\Delta_{1,d-1}$ is non-zero and given as
follows: $$C_i \cdot \Delta_{i,d-i} = 1, \ \ C_i \cdot \Delta_{1, d-1}
= i+1.$$ Finally, the intersection number of $C_i$ with all the other
boundary divisors is zero.  When $i=1$, we have to modify the
intersection number of $C_1$ with $\Delta_{1,d-1}$ to read $C_1 \cdot
\Delta_{1,d-1} = 3$. Next consider the one-parameter family $B_1$ of
rational curves of degree $d$ that contain $d+2$ general points and
intersect a general line. The intersection number of $B_1$ with all
the boundary divisors but $\Delta_{1,d-1}$ is zero. Clearly $B_1 \cdot
D_{\text{deg}} = 0$. By the algorithm for counting rational curves in
projective space given in \cite{vakil:rationalelliptic} it follows
that $$B_1 \cdot \mc{H} = \frac{d^2 + d -2}{2}, \ \ \ B_1 \cdot
\Delta_{1,d-1} = \frac{(d+2)(d+1)}{2}.$$ This determines the class of
$D_{\text{deg}}$ up to a constant multiple. In order to determine the
multiple, consider the curve $C$ that consists of a fixed degree $d-1$
curve and a pencil of lines in a general plane intersecting the curve
in one point. The curve $C$ has intersection number zero with all the
boundary divisors but $\Delta_{1,d-1}$ and has the following
intersection numbers:
$$C \cdot \mc{H} =1, \ \ C \cdot D_{\text{deg}} = 1, \ \ C \cdot
\Delta_{1,d-1} = -1.$$ The lemma follows from these intersection
numbers.
\end{proof}


Consider the one-parameter family $B_k$ of rational curves of degree
$d$ in $\PP^d$ that contain $d+2$ general fixed points and intersect a
general linear space $\PP^k$ and a general linear space $\PP^{d-k}$
for $1 \leq k \leq \lfloor d/2 \rfloor$. When $k=1$, we omit the linear space
$\PP^{d-1}$.  A general member of $B_k$ is a rational normal
curve. This follows, for example, from Lemma 14 of
\cite{fultonpan:kontsevich}. By Lemma \ref{moving} it follows that
$C_k$ is a moving curve for every $k$. The only reducible elements of
$C_1$ are unions of curves of degree 1 and $d-1$. For $k > 1$, the
only reducible curves contained in $C_k$ have degrees $(1, d-1)$ or
$(k, d-k)$. Since the $d+2$ points always span $\PP^d$, $B_k \cdot
D_{\text{deg}} = 0$ for every $k$. It follows that the moving curves
$C_k$ span the null-space of $D_{\text{deg}}$ in the cone of
curves. Observe that these curves give a proof of Theorem \ref{slope}.
\medskip

\begin{proof}[Proof of Theorem~\ref{slope}] Since $D_{\text{deg}}$ is
  an effective divisor class with slope $\frac{1}{d+1}$, the slope of
  $\text{Eff}_{d,d}$ is at least $\frac{1}{d+1}$. On the other hand,
  there are moving curves that have intersection number zero with
  $D_{\text{deg}}$. It follows that the slope of $\text{Eff}_{d,d}$ is
  at most $\frac{1}{d+1}$.
 \end{proof}

\subsection{Open problem about the slope of $\overline{M}_g$} Recall
  that the slope $s(g)$ of $\overline{M}_g$ is defined by $$s(g) :=
  \inf_{\alpha} \{ \ \alpha \lambda - \delta \ \text{ is on the
  effective cone} \ \},$$ where $\lambda$ is the Hodge class and
  $\delta$ is the total boundary class. The slope of the moduli space
  of stable curves $\overline{M}_g$ is non-negative (see
  \cite{joe:slope}). Currently, all known effective divisors on
  $\overline{M}_g$ have slope greater than $6$. As the genus tends to
  infinity the slopes of the Brill-Noether divisors tend to 6 from
  above. Determining the slope, even giving a positive lower bound for
  it, is an important problem with applications to problems such as
  the Schottky problem and the Kodaira dimension of
  $\overline{M}_g$. One can give lower bounds on the slope by
  producing moving curves on $\overline{M}_g$. To the best of our
  knowledge, currently known moving curves give bounds on the slope
  that tend to zero with the genus. \smallskip

The proof of Theorem \ref{slope} suggests a family of moving curves
that might improve known bounds. Let $C_g$ be the one-parameter family
of canonical curves in $\PP^{g-1}$ that contain the maximum possible
number of general points and intersect general linear spaces such that
the sum of the codimensions of all these linear spaces (including the
points) add up to $g^2 + 3g -5$ plus the number of linear spaces. When
$ g \geq 8$, this amounts to considering canonical curves that contain
$g+5$ general points and intersect a general $\PP^{g-7}$. \smallskip

For example, $C_3$ is the one-parameter family of genus 3 canonical
curves that contain 13 general points in $\PP^2$. It has intersection
number zero with the divisor of hyperelliptic curves. $C_4$ is the
one-parameter family of genus 4 canonical curves that contain 9
general points and intersect 5 general lines in $\PP^3$. It has
intersection number zero with the Petri divisor of curves whose
canonical image lies on a singular quadric. $C_5$ is the one-parameter
family of genus 5 canonical curves in $\PP^4$ that contain 11 general
points and intersect a general line.  It has intersection number zero
with the Brill-Noether divisor of trigonal curves. $C_6$ is the
one-parameter family of genus 6 canonical curves in $\PP^5$ that
contain 11 general points and intersect a general line and a general
plane. It has intersection number zero with the Petri divisor of
curves that lie on a singular quintic Del Pezzo surface. When $g \leq
6$, the curves $C_g$ give the sharp slope bound. The analogy with
rational curves and these small-genus examples suggest that this
family is well-worth studying. Unfortunately we do not know the
intersection of $C_g$ with the classes $\lambda$ and $\delta$ in
general. It would be interesting to determine these intersections.




\section{The effective cone of $\Kgnb{0,0}(\PP^d,d)$}\label{proof}

In this section we prove that every effective divisor class in
$\Kgnb{0,0}(\PP^d,d)$ is a positive linear combination of
$D_{\text{deg}}$ and the boundary divisors. This proves Theorem
\ref{mainthm}. \smallskip

Since $D_{\text{deg}}$ and the boundary divisors are effective, any
positive linear combination also is a class in the effective cone. In
order to prove Theorem \ref{mainthm} we have to show that we can write
the class of every effective divisor as $$\alpha D_{\text{deg}} +
\sum_{k=1}^{\lfloor d/2 \rfloor}\beta_{k,d-k} \ \Delta_{k,d-k}, $$ where
$\alpha$ and $\beta_{k,d-k}$ are non-negative.
\smallskip

First, observe that if $D$ is an effective divisor on
$\Kgnb{0,0}(\PP^d,d)$ and $D$ has the class $$a \mc{H} +
\sum_{k=1}^{\lfloor d/2 \rfloor}b_{k,d-k} \Delta_{k,d-k}, $$ then $a
\geq 0$. Furthermore, if $a = 0$, then $b_{k,d-k} \geq 0$. Consider
a general projection of the $d$-th Veronese embedding of $\PP^2$ to
$\PP^d$. Consider the image of a pencil of lines in $\PP^2$. By Lemma
\ref{moving} this is a moving one-parameter family $C$ of degree $d$
rational curves that has intersection number zero with the boundary
divisors. It follows from the inequality $C \cdot D \geq 0$ that $a
\geq 0$. \smallskip

Furthermore, suppose that $a=0$. Consider a general pencil of $(1,1)$
curves on $\PP^1 \times \PP^1$. Take a general projection to $\PP^d$
of the embedding of $\PP^1 \times \PP^1$ by the linear system
$\OO_{\PP^1 \times \PP^1}(i,d-i)$. By Lemma \ref{moving} the image of
the pencil gives a moving one-parameter family $C$ of degree $d$
curves whose intersection with $\Delta_{k,d-k}$ is zero unless
$k=i$. The relation $C \cdot D \geq 0$ implies that if $a=0$, then
$b_{i,d-i} \geq 0$. We conclude that Theorem \ref{mainthm} is true if
$a=0$. We can, therefore, assume that $a > 0$.  \smallskip

Suppose that for every $1 \leq i \leq \lfloor d/2 \rfloor$, we could
construct a moving curve $C_i$ in $\Kgnb{0,0}(\PP^d,d)$ with the
property that $C_i \cdot \Delta_{k,d-k} = 0$ for $k \not= i$ and that
the ratio of $C_i \cdot \Delta_{i,d-i}$ to $C_i \cdot \mc H$ is given
by
\begin{equation}\label{ratio}
\frac{C_i \cdot \Delta_{i,d-i}}{C_i \cdot \mc{H}} =
\frac{d+1}{i(d-i)}.
\end{equation} Observe that given these intersection numbers,
Lemma \ref{theclass} implies that $C_i \cdot D_{\text{deg}} =
0$. Theorem \ref{mainthm} follows from the inequalities $C_i \cdot D
\geq 0$. \smallskip


In the rest of this section we will first give a construction of
one-parameter families of $C_i$ with these properties. However, our
construction will depend on the Harbourne-Hirschowitz conjecture. We
will then modify the construction to get a sequence of curves (not
depending on any conjectures) that ``approximate'' these intersection
numbers. These curves will suffice to conclude Theorem \ref{mainthm}.


\subsection{Construction 1, depending on the Harbourne-Hirschowitz
  conjecture} Let $F_1$ and $F_2$ denote the two fiber classes on
  $\PP^1 \times \PP^1$. We will abuse notation and denote the proper
  transform of the fibers in any blow-up of $\PP^1 \times \PP^1$ also
  by $F_1$ and $F_2$. Let $d, j$ and $k$ be positive integers subject
  to the condition that $ 2k \leq d$. Consider $S$ the blow-up of
  $\PP^1 \times \PP^1$ in $j(d+1)$ general points $p_1, \dots,
  p_{j(d+1)}$. Let $E_i$ denote the $i$-th exceptional divisor lying
  over $p_i$. Let $L(j)$ be the following linear system on $S$: $$L(j) = d \
  F_1 + \frac{jk(k+1)}{2} \ F_2 - \sum_{i=1}^{j(d+1)} k \ E_i.$$

Suppose $M$ is a linear system on $S$ and that $M-F_2$ is non-special,
that is $$h^1 (S, \OO_S(M-F_2)) = 0.$$ Consider the exact sequence $$0
\rightarrow \OO_S(M-F_2) \rightarrow \OO_S(M) \rightarrow \OO_{F_2}(M)
\rightarrow 0.$$ The long exact sequence of cohomology implies that
taking the one-parameter family of proper transforms of the fiber
class $F_2$ under the image of the linear system $|M|$ gives a
one-parameter family of rational curves of degree $M \cdot F_2$
spanning $\PP^{M \cdot F_2}$.  \smallskip

In particular, suppose that $L(j)-F_2$ is non-special. Then by the
discussion in the previous paragraph, the linear system $L(j)$ embeds
the general curve in the linear system $|F_2|$ on $S$ as a rational
normal curve of degree $d$ in $\PP^d$. We thus obtain a one-parameter
family $C_k(j)$ that has intersection number zero with all the
boundary classes except for $\Delta_{k,d-k}$. Moreover, $C_k(j) \cdot
D_{\text{deg}} = 0$. Hence, Theorem \ref{mainthm} would immediately
follow if $L(j) - F_2$ were non-special for at least one value of
$j$. \smallskip


We recall that the celebrated conjecture due to Harbourne and
Hirschowitz characterizes the linear systems that are special on a
general blow-up of $\PP^2$ as those linear systems that have a
multiple $(-1)$-curve in their base locus. Here we will need a weaker
form of the conjecture (see \cite{ciliberto:-1}).

\begin{conjecture}[Harbourne-Hirschowitz]\label{harhir}
Let $M$ be a complete linear system on a general blow-up $S$ of
$\PP^2$. If $E \cdot M$ is non-negative for every $(-1)$-curve $E$ on $S$,
then $M$ is non-special.
\end{conjecture}
Since the blow-up of $\PP^1 \times \PP^1$ at a point is isomorphic to
the blow-up of $\PP^2$ at three points, the Harbourne-Hirschowitz
Conjecture applies to the linear systems $L(j)$.
The class of any $(-1)$-curve on $S$ may be expressed as

$$\alpha F_1 + \beta F_2 - \sum_{i=1}^{j(d+1)} \gamma_i E_i,$$ where
$\alpha, \beta$ and $\gamma_i$ are non-negative integers.
Since for a $(-1)$-curve $E$ we have $E \cdot K = -1$, it follows
that $$- \sum_{i=1}^{j(d+1)} \gamma_i = 1 - 2\alpha - 2\beta.$$
The intersection of the $(-1)$-curve with $L(j)$ is 

\begin{align*}
 d\alpha + \beta \left( \frac{jk(k+1)}{2} -1 \right) - k
\sum_{i=1}^{j(d+1)} \gamma_i =  (d-2k)
\beta + \alpha \left( \frac{jk(k+1)}{2}-2k -1 \right) + k.
\end{align*}
 When $k>3$ and $j \geq 1$; or $k=2, 3$ and $j > 1$; or $k=1$ and $j
 \geq 3$, the intersection is non-negative. We conclude the following:

\begin{proposition}\label{HH}
Suppose Conjecture \ref{harhir} holds for $L(j)- F_2$ for some $j \geq
1$.  Then the effective cone of $\Kgnb{0,0}(\PP^d,d)$ is spanned by
the classes of $D_{\text{deg}}$ and the boundary
divisors. Furthermore, every codimension one face of the effective
cone is the null-locus of a moving curve.
\end{proposition}

\begin{remark}
It is easy to prove that $L(j) - F_2$ is non-special for small values
of $d$ and $k$ and to deduce Proposition \ref{HH} without any
conditions. However, we could not see how to prove the non-specialty
of $L(j) - F_2$ in general.
\end{remark}

\subsection{Construction 2, completing the proof} We modify the
previous construction by imposing fewer $k$-fold points on the linear
system $d \ F_1 + \frac{jk(k+1)}{2} F_2 $ on $\PP^1 \times \PP^1$. If
we do not impose too many $k$-fold points on the linear system, we can
prove the non-specialty of the desired linear system. The following
proposition makes this precise.

\begin{proposition}\label{asymp}
Let $k,j$ and $d$ be positive integers subject to the condition that
$2k \leq d$.  There exists an integer $n(k,d)$ depending only on $k$
and $d$ such that the linear system
$$L'(j) = d \ F_1 + \left( \frac{jk(k+1)}{2} - 1 \right) \ F_2 -
  \sum_{i=1}^{j(d+1)- n(k,d)} k \ E_i - \sum_{i=j(d+1)- n(k,d) +
  1}^{j(d+1) + n(k,d)\frac{(k-1)(k+2)}{2}} E_i $$ on the blow-up of
  $\PP^1 \times \PP^1$ at $j(d+1) + n(k,d)\frac{(k-1)(k+2)}{2}$
  general points is non-special for every $j >> 0$. The integer
  $n(k,d)$ may be taken to be
$$n(k,d)= \lceil 2(d+1)/k \rceil.$$
\end{proposition}

 Proposition \ref{asymp} implies Theorem \ref{mainthm}. As in the
 previous subsection we consider the blow-up of $\PP^1 \times
 \PP^1$ in $$j(d+1) + \frac{n(k,d)(k-1)(k+2)}{2}$$ general points. The proper
 transform of the fibers $F_2$ under the linear system 
$$L'(j) = d \ F_1 + \frac{jk(k+1)}{2} \ F_2 - \sum_{i=1}^{j(d+1)-
  n(k,d)} k \ E_i - \sum_{i=j(d+1)- n(k,d) + 1}^{j(d+1) +
  n(k,d)\frac{(k-1)(k+2)}{2}} E_i $$ gives a one-parameter family
  $C_k(j)$ of rational curves of degree $d$ that has intersection
  number zero with $D_{\text{deg}}$.  Letting $j$ tend to infinity we
  obtain a sequence of moving curves $C_k(j)$ in $\Kgnb{0,0}(\PP^d,d)$
  that has intersection zero with all the boundary divisors but
  $\Delta_{1,d-1}$ and $\Delta_{k,d-k}$. Unfortunately, the
  intersection of $C_k(j)$ with $\Delta_{1,d-1}$ is not zero and the
  ratio of $C_k(j) \cdot \mc{H}$ to $C_k(j) \cdot \Delta_{k,d-k}$ is
  not the one required by Equation (\ref{ratio}). However, as $j$
  tends to infinity, the ratio of the intersection numbers $C_k(j)
  \cdot \Delta_{1,d-1}$ to $C_{k}(j) \cdot \mc{H}$ tends to zero and
  the ratio of $C_k(j) \cdot \Delta_{k,d-k}$ to $C_k(j) \cdot
  \mc{H}$ tends to the desired ratio
  $\frac{d+1}{k(d-k)}$. Theorem \ref{mainthm} follows.  \smallskip


\begin{proof}[Proof of Proposition \ref{asymp}] 
The specialization technique developed in \S 2 of
\cite{stephanie:game} yields the proof of the proposition. We will
specialize the points of multiplicity $k$ one by one onto a point
$q$. At each stage the $k$-fold point that we specialize will be in
general position. We will first slide the point along a fiber $f_1$ in
the class $F_1$ onto the fiber $f_2$ in the fiber class $F_2$
containing the point $q$. We then slide the point onto $q$ along
$f_2$. We will record the flat limit of this degeneration. \smallskip

There is a simple checker game that describes the limits of these
degenerations. This checker game for $\PP^2$ is described in \S 2 of
\cite{stephanie:game}. The details for $\PP^1 \times \PP^1$ are
identical. The global sections of the linear system $\OO_{\PP^1 \times
\PP^1} (a,b)$ are bi-homogeneous polynomials of bi-degree $a$ and $b$
in the variables $x,y$ and $z,w$, respectively. A basis for the space
of global sections is given by $x^iy^{a-i}z^j w^{b-j}$, where $0 \leq
i \leq a$ and $0 \leq j \leq b$. We can record these monomials in a
rectangular $(a+1) \times (b+1)$ grid. In this grid the box in the
$i$-th row and the  $j$-th column corresponds to the monomial
$x^iy^{a-i}z^j w^{b-j}$.


\begin{figure}[htbp]
\begin{center}
\epsfig{figure=initch.eps}
\end{center}
\caption{Imposing a triple point on $\OO_{\PP^1 \times
\PP^1} (4,6)$.}
\label{initch}
\end{figure}


If we impose an ordinary $k$-fold point on the linear system at
$([x:y], [z:w]) = ([0:1],[0:1])$, then the coefficients of the
monomials $$y^aw^b, xy^{a-1}w^b, \dots,
x^{k-1}y^{a-k+1}z^{k-1}w^{b-k+1}$$ must vanish. We depict this by
filling in a $k \times k$ triangle of checkers into the boxes at the
upper left hand corner as in Figure \ref{initch}. The coefficients of
the monomials represented by boxes that have checkers in them must
vanish. \smallskip


We first slide the $k$-fold point along the fiber $f_1$ onto the point
$([x:y], [z:w]) = ([1:0],[0:1])$. This correspond to the degeneration
$$([x:y], [z:w]) \mapsto ([x:ty], [z:w]).$$ The flat limit of this
degeneration is described by the vanishing of the coefficients of
certain monomials (assuming none of the checkers fall out of the
rectangle). The monomials whose coefficients must vanish are those
that correspond to boxes with checkers in them when we let the
checkers fall according to the force of gravity. The first two panels
in Figure \ref{grav} depict the result of applying this procedure to
a 4-fold point when there is an aligned ideal condition at the point
$([x:y], [z:w]) = ([1:0],[1:0])$. \smallskip

We then follow this degeneration with a degeneration that specializes
the $k$-fold point to $q$ by sliding along the fiber $f_2$. This
degeneration is explicitly given by $$([x:y], [z:w]) \mapsto ([x:y],
[z:tw]).$$ The flat limit is described by the vanishing of the
coefficients of the monomials that have checkers in them when we slide
all the checkers as far right as possible. The last two panels of
Figure \ref{grav} depict this degeneration.

\begin{figure}[htbp]
\begin{center}
\input{check.pstex_t}
\end{center}
\caption{Depicting the degenerations by checkers.}
\label{grav}
\end{figure}   

S. Yang proves that, provided none of the checkers fall out of the
ambient rectangle during these moves, these checker movements do
correspond to the flat limits of the linear systems under the given
degenerations. If one can play this checker game with all the multiple
points that one imposes on a linear system so that during the game
none of the checkers fall out of the rectangle, one can conclude that
the multiple points impose independent conditions on the linear
system. The limit linear system has the expected dimension. In
particular, it is non-special. By upper semi-continuity the original
linear system must also have the expected dimension and be
non-special.  Unfortunately, when one plays this game, occasionally
checkers may fall out of the rectangle. In that case we loose
information on what the limits are. This may happen even if the
original linear system has the expected dimension. \smallskip

In order to conclude the proposition we need to show that if we impose
 at most $j(d+1) - n(k,d)$ points of multiplicity $k$ on the linear
 system $\OO_{\PP^1 \times \PP^1} (d,jk(k+1)/2)$ where $2k \leq d$, we
 do not lose any checkers when we specialize all the $k$-fold points
 by the degeneration just described. This suffices to conclude the
 proposition because general simple points always impose independent
 conditions. \smallskip


The main observation is that if there is a safety net of empty boxes
at the top of the rectangle, then the checkers will not fall out of
the box. The proof of the proposition is completed by noting the
following simple facts.

\begin{enumerate}
\item At any stage of the degeneration the height of the checkers in
  the rectangle is at most $k$ larger than the highest row full of
  checkers.

\item The left most checker of a row is to the lower left of the left
  most checker of any row above it.
\end{enumerate}

If there are at least $(k+1) (d+1)$ empty boxes in our rectangle, then
by the above two observations when we specialize a $k$-fold point we
do not lose any of the checkers. As long as $n(k,d) \geq \lceil
2(d+1)/k \rceil$, there is always at least $(k+1)(d+1)$ boxes
empty. Hence until the stage where we specialize the last $k$-fold
point we cannot lose any checkers. This concludes the proof.
\end{proof}

\begin{remark}
While the asymptotic approach gives a proof of Theorem \ref{mainthm}
independent of the Harbourne-Hirschowitz conjecture, it does not
construct a moving curve that is dual to the codimension one faces of
the effective cone of $\Kgnb{0,0}(\PP^d,d)$.
\end{remark}





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