Tau function and moduli of spin curves

TL;DR

This paper provides an analytic proof of Farkas's formula for spinor divisors in moduli spaces of odd spin curves using the Bergman tau function, and derives theta-null formulas for even spin curves from classical theta function theory.

Contribution

It offers a new analytic proof of Farkas's divisor class formula and connects it with classical theta function theory for even spin curves.

Findings

Analytic proof of Farkas's formula using Bergman tau function

Derivation of theta-null formula from classical theta functions

Enhanced understanding of moduli space of spin curves

Abstract

The goal of the paper is to give an analytic proof of the formula of G. Farkas for the divisor class of spinors with multiple zeros in the moduli space of odd spin curves. We make use of the technique developed by Korotkin and Zograf that is based on properties of the Bergman tau function. We also show how the Farkas formula for the {\it theta-null} in the rational Picard group of the moduli space of even spin curves can be derived from classical theory of theta functions.