Towards the ample cone of \overline{}_{,}
Towards the ample cone of \overline{}_{,}
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朝向 overline{}_{,} 的宽敞圆锥体
DOI：
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发表时间：
2000
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通讯作者：
I. Morrison
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文献类型：
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作者：
A. Gibney;S. Keel;I. Morrison
本文研究亏格$g$的稳定$n点曲线的模空间$\mgn$的充分锥。我们的猜想是，$\mgn$上的一个因子是充分的当且仅当它与所有一维地层(至少有$3g+n-2$个结点的曲线轨迹的分量)有正交。这转化为用线性不等式对锥体的一种简单的猜想描述，并且，由于所有的1-层都是有理的，包括这样的猜想：Mori锥是多面体的，由有理曲线生成。我们的主要结果是猜想成立当且仅当$g=0$成立。更准确地说，存在一个自然的有限映射$r：\vmgn 0。2G+n.\to\mgn$，其图像是所有分量为有理的曲线的轨迹$\rgn$。我们证明了一个因子$D$是NEF当且仅当$D\CDOT E\geq 0$和$r^*(D)$是NEF。我们还给出了$ggeq1$的$MGN$的压缩(即具有连通纤维到射影簇的态射)的结果，表明了任何纤维因子都是通过重言式的(通过忘点给出的)，并且任何双态收缩的例外轨迹都包含在边界内。最后，通过更多的即席论证，证明了某些特殊类的整洁性。
In this paper we study the ample cone of the moduli space $\mgn$ of stable $n$-pointed curves of genus $g$. Our motivating conjecture is that a divisor on $\mgn$ is ample iff it has positive intersection with all 1-dimensional strata (the components of the locus of curves with at least $3g+n-2$ nodes). This translates into a simple conjectural description of the cone by linear inequalities, and, as all the 1-strata are rational, includes the conjecture that the Mori cone is polyhedral and generated by rational curves. Our main result is that the conjecture holds iff it holds for $g=0$. More precisely, there is a natural finite map $r: \vmgn 0. 2g+n. \to \mgn$ whose image is the locus $\rgn$ of curves with all components rational. Any 1-strata either lies in $\rgn$ or is numerically equivalent to a family $E$ of elliptic tails and we show that a divisor $D$ is nef iff $D \cdot E \geq 0$ and $r^*(D)$ is nef. We also give results on contractions (i.e. morphisms with connected fibers to projective varieties) of $\mgn$ for $g \geq 1$ showing that any fibration factors through a tautological one (given by forgetting points) and that the exceptional locus of any birational contraction is contained in the boundary. Finally, by more ad-hoc arguments, we prove the nefness of certain special classes.