The Birational Geometry of Moduli Spaces of Curves

曲线的模空间是一种代数变体（更准确地说是一种格式或一堆），其点参数化g属代数曲线的同构类。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，或在自然投影闭包（称为Deligne-Mumford紧化）中，其点对应于最多有节点奇点的曲线是有用的。边界的分量是稳定n点曲线模空间的映射图像。事实上，这些曲线的模空间的二分几何经常揭示曲线族的有趣性质。研究射影变体的二分几何最有效的方法之一是研究其上的除数和曲线。特别是，如何描述除数的有效锥和非锥，从而描述这些空间的曲线的有效锥，是一个基本问题。要进行的研究计划是对这个问题的多方面的攻击。目的是进一步澄清这些锥体的性质，它们之间的关系，并提供一系列丰富的例子，这些例子将加深对模空间的两族几何的理解，并拓宽关于曲线锥体和除数锥体的一般知识。

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.