Admissible covers and stable maps

TL;DR

This paper establishes a cycle-theoretic formula linking admissible covers and stable maps, unifying and refining existing formulas like the Gromov-Witten/Hurwitz correspondence, with computational verification in low-genus cases.

Contribution

It introduces a new cycle formula that generalizes and refines key existing results in the intersection theory of moduli spaces.

Findings

The formula recovers the Ekedahl-Lando-Shapiro-Vainshtein formula.

It provides a cycle-theoretic refinement of the Gromov-Witten/Hurwitz correspondence.

Computational checks confirm the formula in low-genus scenarios.

Abstract

Moduli spaces of admissible covers and stable maps of target curves give rise to cycles on $\overline{M}_{g,n}$. We prove a formula relating these cycles. It recovers both the Ekedahl-Lando-Shapiro-Vainshtein formula and the Gromov-Witten/Hurwitz correspondence, providing a cycle-theoretic refinement thereof. The formula is verified computationally in low-genus cases with the help of Johannes Schmitt.