On the birational geometry of the moduli space of cyclic branched covers of curves

• Applicant: Dr. Irene Schwarz
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 456444857

In algebraic geometry the Hurwitz scheme usually refers to the moduli space parametrizing simple branched coverings of a given degree d of the projective line by algebraic curves. More generally, Hurwitz schemes can be defined as the moduli spaces parametrizing, for a fixed group G, branched G-coverings C to D of a given degree d of algebraic curves, where C has genus g and D has genus h. These moduli spaces are closely related to Gromov-Witten theory with applications e.g. in string theory. They are, however, rarely studied for their own sake,although they already play a prominent role in the seminal paper of Harris-Mumford proving that the moduli space of algebraic curves has maximal Kodaira dimension (i.e. it is of general type) for (odd) g greater 23.In this research proposal we suggest studying the birational geometry of such Hurwitz schemes and in particular their Kodaira dimension. There are several birational properties such as rationality, unirationality, rational connectedness and uniruledness which all imply negative, i.e. minimal, Kodaira dimension. We, however, will focus on the other extreme of maximal Kodaira dimension which means being of general type.A seminal result of Harris and Mumford establishes this for the related moduli space of stable genus g curves if g is greater than 23; this has been sharpened and generalized in subsequent work, in particular including additional marked points. In my doctoral thesis I have extended the theory to some natural subspaces of the moduli space of genus g curves, in particular the moduli space of nodal curves and the moduli space of hyperelliptic curves of genus g, with additional marked points. To obtain results for Hurwitz schemes, a careful study of their singularities and the calculation of their canonical classes are essential. We also believe that some considerations of Brill-Noether theory could be interesting. As it is unlikely that a uniform result can be found for all branched covers of curves, some simplifying conditions should be imposed. Thus we intend to restrict ourselves to the case of branched cyclic Galois covers of curves over the complex numbers.Furthermore, in a second step, we want to gradually turn our focus on some aspects of enumerative geometry. This subject has been deeply influenced by Gromov-Witten theory and Hurwitz schemes already have made a prominent appearance. Here we hope to profit substantially from the experience and expertise of Prof. Pandharipande's group at the ETH Zurich.

• DFG Programme: WBP Fellowship
• International Connection: Switzerland
• Host: Professor Dr. Rahul Pandharipande