Moduli Spaces of Curves and their Cohomology

在本项目中，将研究以下几个问题。(A)亏格g的非奇异曲线的模空间M(G)是否包含g-2维的完备子簇？它的Deligne-Mumford紧凑化是否包含不与某些边界成分相交的大型完整亚种？(B)上同调的哪一部分来自所谓的重言式类，以及理解这些类之间的关系的问题。(C)有动机地理解曲线的模空间的完全上同调的问题(对于低亏格)。(D)定性地理解出现在点曲线的模空间的上同调中的对称群的表示的问题(点是有序的，它产生对称群的自然作用)。曲线的模空间在许多数学领域和理论物理(弦理论)中起着重要的作用。通过对上述问题的研究，将加深我们对模空间本身、曲线族的理解，并最终加深对任意空间中曲线的性质的理解。人们通常从研究行为良好(即，投射的、非奇异的、连通的)并且由系数为复数(或另一个代数闭域的元素)的方程给出的曲线开始。这种曲线的基本不变量是它的所谓亏格，即一个非负整数。如果画一条复杂曲线的真实图画，就会看到一个(紧凑且连通的)曲面；它的亏格是“洞”的数量。区分不同亏格的非同构曲线和刻画给定亏格g的曲线的同构类往往是很重要的，亏格g的曲线的模空间M(G)是这样一个空间，它的点对应于这些同构类，并且它的性质是基上的亏格曲线族带有从基到M(G)的自然映射。曲线的模空间不仅出现在数学的许多分支中，也出现在理论物理中。人们对它进行了深入的研究，并有充分的理由。例如，关于M(G)的结果告诉我们一些关于曲线族的事情，从而最终关于代数几何中的任意解空间。

In this project, the following problems will be studied. (A) Does M(g), the moduli space of nonsingular curves of genus g, contain complete subvarieties of dimension g-2? Does its Deligne-Mumford compactification contain large complete subvarieties not intersecting certain boundary components? (B) The question which part of the cohomology comes from the so-called tautological classes and the problem of understanding the relations between these classes. (C) The problem of understanding the entire cohomology of moduli spaces of curves motivically (for low genus). (D) The problem of obtaining a qualitative understanding of the representations of the symmetric group appearing in the cohomology of moduli spaces of pointed curves (the points are ordered, which yields a natural action of the symmetric group). The moduli space of curves plays a fundamental role in many areas of mathematics and in theoretical physics (string theory). Results obtained by studying the problems above will deepen our understanding of the moduli space itself, of families of curves, and ultimately of the fibrations in curves of arbitrary spaces. One usually begins by studying curves that are well-behaved (i.e., projective, nonsingular, connected) and that are given by equations whose coefficients are complex numbers (or elements of another algebraically closed field). The fundamental invariant of such a curve is its so-called genus, a nonnegative integer. If one draws a real picture of a complex curve, one sees a (compact and connected) surface; its genus is the number of `holes'. Often it is important to distinguish non-isomorphic curves of the same genus and to describe the isomorphism classes of curves of a given genus g. The moduli space M(g) of curves of genus g is a space whose points correspond to these isomorphism classes and it has the property that a family of curves of genus g over a base comes with a natural map from the base to M(g). The moduli space of curves makes its appearance in many branches of mathematics and also in theoretical physics. It is studied intensively and for good reasons. E.g., results about M(g) tell us something about families of curves, thus ultimately about arbitrary solution spaces in algebraic geometry.