Moduli Spaces of Curves and their Cohomology

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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.