Tau functions, Prym-Tyurin classes and loci of degenerate differentials

TL;DR

This paper investigates the structure of the Picard group of the moduli space of holomorphic n-differentials, introducing Prym-Tyurin classes and expressing them in terms of boundary divisors through analytic and algebraic methods.

Contribution

It defines Prym-Tyurin classes in the Picard group of the moduli space and provides two proofs expressing these classes as linear combinations of boundary divisors.

Findings

Prym-Tyurin classes are expressed as linear combinations of boundary divisors.

Two different proofs are provided: an analytic approach using the Bergman tau function.

An algebraic approach involving cohomological computations is also developed.

Abstract

We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve.