The H-tautological ring

TL;DR

This paper introduces the H-tautological ring, extending tautological classes to moduli spaces of admissible Galois covers, including new classes and computational methods for intersection theory.

Contribution

It develops the theory of H-tautological rings with restriction-corestriction morphisms and provides algorithms for intersection computations, expanding the scope of tautological classes.

Findings

Defined additive generators for H-tautological rings

Developed an algorithm for intersection computations

Applied to compute integrals of Harris-Mumford loci

Abstract

We extend the theory of tautological classes on moduli spaces of stable curves to the more general setting of moduli spaces of admissible Galois covers of curves, introducing the so-called H-tautological ring. The main new feature is the existence of restriction-corestriction morphisms remembering intermediate quotients of Galois covers, which are a rich source of new classes. In particular, our new framework includes classes of Harris-Mumford admissible covers on moduli spaces of curves, which are known in some (and speculatively many more) examples to lie outside the usual tautological ring. We give additive generators for the H-tautological ring and show that their intersections may be algorithmically computed, building on work of Schmitt-van Zelm. As an application, we give a method for computing integrals of Harris-Mumford loci against tautological classes of complementary dimension, recovering and giving a mild generalization of a recent quasi-modularity result of the author for covers of elliptic curves.