Moduli Spaces of Curves and their Cohomology

在本项目中，将研究以下问题。(A)亏格为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紧化是否包含不与某些边界分支相交的大的完全子簇？(B)上同调的哪一部分来自所谓重言式类的问题，以及理解这些类之间关系的问题。(C)曲线模空间的全上同调的动力学理解问题（低亏格）。(D)获得定性理解对称群的表示出现在尖曲线的模空间的上同调中的问题（点是有序的，这产生了对称群的自然作用）。 曲线的模空间在数学和理论物理（弦理论）的许多领域中起着基础性的作用。研究上述问题所得到的结果将加深我们对模空间本身、曲线族以及任意空间中曲线的纤维化的理解。人们通常从研究表现良好的曲线开始（即，投射的，非奇异的，连通的），并且由系数为复数的方程（或另一个代数闭域的元素）给出。这样一条曲线的基本不变量是它所谓的亏格，一个非负整数。如果我们画一幅复曲线的真实的图，我们看到的是一个（紧致的和连通的）曲面;它的亏格就是“洞”的个数。通常区分同一亏格的非同构曲线和描述给定亏格g的曲线的同构类是很重要的。亏格g的曲线的模空间M（g）是其点对应于这些同构类的空间，并且它具有这样的性质：在基上亏格g的曲线族具有从基到M（g）的自然映射。曲线的模空间出现在数学的许多分支中，也出现在理论物理中。它被深入研究，并有很好的理由。例如，在一个示例中，关于M（g）的结果告诉我们有关曲线族的一些信息，从而最终告诉我们有关代数几何中任意解空间的一些信息。

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.