Counting differentials with fixed residues

TL;DR

This paper provides a comprehensive intersection-theoretic approach to counting meromorphic differentials on the Riemann sphere with fixed residues, extending previous geometric and polynomial map methods.

Contribution

It introduces a novel intersection theory framework on moduli spaces to solve the counting problem for differentials with arbitrary residues.

Findings

Complete solution to the counting problem with fixed residues.

Identification of combinatorial properties of the solution formula.

Extension of previous methods to arbitrary residue conditions.

Abstract

We investigate the count of meromorphic differentials on the Riemann sphere possessing 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 residue conditions and provide a complete solution to this problem. We also determine interesting combinatorial properties of the solution formula.