Tautological classes on the moduli space of hyperelliptic curves with rational tails

TL;DR

This paper provides a comprehensive description of tautological classes and relations on the moduli space of hyperelliptic curves with rational tails, linking algebraic and cohomological perspectives.

Contribution

It introduces a complete characterization of tautological relations using Jacobian comparisons, confirming all relations originate from the Jacobian side.

Findings

All tautological relations derive from the Jacobian approach.

Intersection pairings are proven to be perfect in all degrees.

Tautological algebra matches its cohomological image, identified with monodromy-invariant classes.

Abstract

We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology. The connection with recent conjectures by Pixton is also discussed.