Szeg\H{o} kernels and Scorza quartics on the moduli space of spin curves

TL;DR

This paper extends the map from the moduli space of stable spin curves to their associated Scorza curves, computes related classes, and offers new interpretations using theta constants, impacting the understanding of superperiod maps and the geometry of spin moduli spaces.

Contribution

It introduces an extension of the Scorza curve map to the moduli space of stable spin curves and provides new geometric and interpretative insights.

Findings

Computed the class of the Szeg ext{"o}-Hodge bundle.

Provided a new interpretation of the Scorza quartic via theta constants.

Established a lower bound for the slope of the moduli space's movable cone.

Abstract

We describe an extension at the level of the moduli space of stable spin curves of genus g of the map associating to an ineffective spin structure its Scorza curve (equivalently, the vanishing locus of its Szeg\H{o} kernel). We compute the class of the Szeg\H{o}-Hodge bundle, then find an unconditional new interpretation, in terms of theta constants, of the Scorza quartic uniquely associated to an even spin structure. Our results describe the superperiod map from the moduli space of supersymmetric curves in the neighborhood of the theta-null divisor and provide a lower bound for the slope of the movable cone of the moduli space of spin curves.