The universal theta divisor over the moduli space of curves

TL;DR

This paper classifies the birational geometry of the universal theta divisor over the moduli space of curves, revealing its rationality for low genus and general type for higher genus, using intersection theory.

Contribution

It provides a complete birational classification of the universal theta divisor and the universal symmetric product over the moduli space of curves.

Findings

Th_g is rational for g<12

Th_g is of general type for g≥12

Intersection theory of the Gauss map's antiramification locus is key

Abstract

We carry out a complete birational classification of the universal theta divisor Th_g over the moduli space of curves of genus g, and show that Th_g enjoys good rationality properties for g<12, and is a variety of general type for g\geq 12. The key ingredient is an intersection-theoretic study of the universal antiramification locus of the Gauss map. We also present a complete classification of the universal symmetric product of degree g-2 over M_g.