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Quadratic Pseudostable Hodge Integrals and Mumford's Relations

Renzo Cavalieri, Matthew M. Williams

TL;DR

This work establishes a precise bridge between quadratic Hodge integrals on moduli spaces of pseudostable curves and their stable counterparts. It proves a pseudostable version of Mumford's relation, expressing the pullback of $\mathbb{E}\oplus\mathbb{E}^\vee$ in boundary/descent terms and providing a recursive framework via decorated dual graphs. It then gives a complete formula for all quadratic pseudostable Hodge integrals in terms of stable integrals, with the generating functions related by $\mathcal{F}^{ps}=e^{z/24}\mathcal{F}$, enabling efficient computation. The results illuminate the combinatorial structure governing pseudostable tautological intersections and suggest pathways to pseudostable Gromov-Witten theory and mirror-symmetry questions. Overall, the paper delivers both explicit identities and generating-function tools that significantly simplify and organize quadratic pseudostable Hodge calculations.

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 $λ$ classes and obtain an explicit comparison of arbitrary pseudostable and stable quadratic Hodge integrals.

Quadratic Pseudostable Hodge Integrals and Mumford's Relations

TL;DR

This work establishes a precise bridge between quadratic Hodge integrals on moduli spaces of pseudostable curves and their stable counterparts. It proves a pseudostable version of Mumford's relation, expressing the pullback of in boundary/descent terms and providing a recursive framework via decorated dual graphs. It then gives a complete formula for all quadratic pseudostable Hodge integrals in terms of stable integrals, with the generating functions related by , enabling efficient computation. The results illuminate the combinatorial structure governing pseudostable tautological intersections and suggest pathways to pseudostable Gromov-Witten theory and mirror-symmetry questions. Overall, the paper delivers both explicit identities and generating-function tools that significantly simplify and organize quadratic pseudostable Hodge calculations.

Abstract

This paper studies the relationship between quadratic Hodge classes on moduli spaces of pseudostable and stable curves given by the contraction morphism While Mumford relations do not hold in the pseudostable case, we show that one can express the (pullback via of the) Chern classes of solely in terms of descendants and strata classes. We organize the combinatorial structure of the pullback of products of two pseudostable classes and obtain an explicit comparison of arbitrary pseudostable and stable quadratic Hodge integrals.
Paper Structure (13 sections, 10 theorems, 122 equations, 4 figures)

This paper contains 13 sections, 10 theorems, 122 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Context and motivation
  4. Structure of the paper
  5. Acknowlegments
  6. Background
  7. The moduli spaces of stable and pseudostable curves
  8. Tautological classes
  9. Dual graphs of boundary strata
  10. Mumford's relation for pseudostable $\lambda$ classes
  11. Applications
  12. Quadratic Pseudostable Hodge integrals
  13. Applications

Key Result

Theorem 2.5

For all pseudostable indices $(g, n)$, we have and

Figures (4)

  • Figure 1: The dual graphs for the strata $\mathcal{G}_*^1(1)$ and $\mathcal{G}_*^2(1)$ in $\overline{\mathcal{M}}_{3, 0}$.
  • Figure 2: The dual graphs for the strata $\mathcal{G}_*^1(1)$ and $\mathcal{G}_*^2(1)$ in $\overline{\mathcal{M}}_{3, 0}$.
  • Figure 3: Up to isomorphism, the two generic $(G, H)$-graphs for $G$ and $H$ in Figure \ref{['fig:2dualG1G2']}.
  • Figure 4: The resulting product of $G$ and $H$.

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: As in PSHI
  • Definition 2.4
  • Theorem 2.5: Theorem 2.4, PSHI
  • Definition 2.6: graberpandharipandeYang
  • Example 2.7
  • Example 2.8
  • Remark 2.9
  • Definition 2.10: based on graberpandharipandeYang
  • ...and 26 more