Powers of the theta divisor and relations in the tautological ring

TL;DR

This paper demonstrates how vanishing properties of the theta divisor influence tautological relations in moduli spaces of curves, linking geometric vanishing conditions to algebraic relations and providing an algorithm for tautological class expressions.

Contribution

It establishes new connections between theta divisor vanishing, double ramification cycle relations, and tautological ring relations, along with an algorithm for tautological class reduction.

Findings

Vanishing of the $(g+1)$-st power of the theta divisor implies tautological relations.

Pixton's double ramification cycle relations imply Graber-Vakil's Theorem $ ext{ extasteriskcentered}$.

Provides an algorithm to express high codimension tautological classes on $ar{ ext{M}}_{g,n}$.

Abstract

We show that the vanishing of the $(g+1)$-st power of the theta divisor in the cohomology and Chow rings of the universal abelian variety implies, by pulling back along a collection of Abel-Jacobi maps, the vanishing results in the tautological ring of $\mathcal{M}_{g,n}$ of Looijenga, Ionel, Graber-Vakil, and Faber-Pandharipande. We also show that Pixton's double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem $\star$ of Graber and Vakil. Moreover, our proof provides an algorithm for expressing any tautological class on $\overline{\mathcal{M}}_{g,n}$ of sufficiently high codimension as a tautological class supported on the boundary.