Quadratic Pseudostable Hodge Integrals and Mumford's Relations

TL;DR

This paper explores the relationship between quadratic Hodge classes on moduli spaces of pseudostable and stable curves, providing explicit formulas and comparisons despite the failure of Mumford relations in the pseudostable case.

Contribution

It introduces a method to express Chern classes of certain bundles in terms of descendants and strata classes on pseudostable curves, and compares pseudostable and stable quadratic Hodge integrals.

Findings

Expressed Chern classes using descendants and strata classes.

Organized combinatorial structure of pullback of pseudostable lambda classes.

Provided explicit comparison of pseudostable and stable quadratic Hodge integrals.

Abstract

This paper studies the relationship between quadratic Hodge classes on moduli spaces of pseudostable and stable curves given by the contraction morphism $\mathcal{T}.$ While Mumford relations do not hold in the pseudostable case, we show that one can express the (pullback via $\mathcal{T}$ of the) Chern classes of $\mathbb{E}\oplus \mathbb{E}^\vee$ solely in terms of descendants and strata classes. We organize the combinatorial structure of the pullback of products of two pseudostable $\lambda$ classes and obtain an explicit comparison of arbitrary pseudostable and stable quadratic Hodge integrals.