Prado Godoy, Miguel Angel. “Counting differentials with fixed residues”, Boston College, 2024. http://hdl.handle.net/2345/bc-ir:109921.

Abstract

We investigate the count of meromorphic differentials on the Riemann sphere pos-sessing a single zero, multiple poles with prescribed orders, and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed-point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residues and pole orders, which provides a complete solution to this problem. We also determine interesting combinatorial properties of the solution formula. This thesis is organized as follows: In Chapter 1 we give an introduction to the problem and summarize the main results obtained. In Chapter 2 we review the compactification of moduli spaces of differentials and introduce various divisor classes. In Section 2.3 we explain how to identify the universal line bundle class with the divisor class of the locus of differentials satisfying a general given residue tuple and prove Theorem 1.0.1 (i). In Section 2.4 we impose exactly one independent partial sum vanishing condition to the residues and prove Theorem 1.0.1 (ii). In Section 2.5 we give a polynomial expression in terms of the zero order for the degree of mixed products between powers of the dual tautological class and the psi-class of the zero. Finally in Chapter 3 we prove Theorem 1.0.2 for arbitrary residues and investigate combinatorial properties of the solution formula. We have also verified our formula numerically for a number of cases by using the software package [CMZ2].