The Birational Geometry of Moduli Spaces of Curves

曲线的模空间是一个代数簇（更准确地说是一个scheme或stack），它的点将亏格g的代数曲线的同构类参数化。 在研究曲线族的行为时，将每条曲线视为模空间中的一个点是有用的，或者在自然射影闭包（称为Deligne-Mumford紧化）中，其点对应于具有最多节点奇点的曲线。边界的分量是稳定n点曲线的模空间映射的象。事实上，这些不同模空间的曲线的双有理几何经常揭示曲线族的有趣性质，研究射影簇的双有理几何最有成效的方法之一是研究它们上的因子和曲线。 特别是，它是一个基本问题，以描述有效的和nef锥的因子，因此有效锥的曲线，这些空间。要进行的研究计划是对这个问题的多方面的研究。 我们的目标是进一步澄清这些锥的性质，它们之间的关系，以及提供一个丰富的例子，这将加深理解的双有理几何的模空间，也扩大了一般知识锥曲线和除数。

DMS-0331315Angela GibneyThe moduli space of curves is an algebraic variety (more accurately ascheme or a stack) whose points parametrize isomorphism classes ofalgebraic curves of genus g. In studying the behavior of families ofcurves, it is useful to consider each curve as a point in the moduli space, or in the natural projective closure (called the Deligne-Mumford compactification) whose points correspond to curves with at most nodal singularities. The components of the boundary are the images of maps from moduli spaces of stable n-pointed curves. In fact, it turns out that the birational geometry of these various moduli spaces of curves frequently reveals interesting properties of families of curves.One of the most fruitful ways to study the birational geometry ofprojective varieties is to investigate the divisors and curves on them. In particular, it is a fundamental problem to describe the effective and nef cones of divisors, and hence the effective cone of curves of these spaces. The research program to be pursued is a many faceted attack on this problem. The goal is to provide further clarification of the nature of these cones, the relationships between them as well as to provide a collection of rich examples which would deepen understanding of the birational geometry of the moduli spaces and also broaden general knowledge about cones of curves and divisors.