Moduli spaces of twisted maps to smooth pairs

TL;DR

This paper investigates moduli spaces of twisted maps to smooth pairs, providing geometric insights into orbifold and logarithmic Gromov--Witten invariants, classifying components via tropical data, and exploring polynomial degree relations in the context of higher genus invariants.

Contribution

It introduces a comprehensive geometric framework for moduli spaces of twisted maps, classifies their components, and establishes polynomial degree relations, extending previous results to higher genus cases.

Findings

Classified irreducible components using tropical data.

Computed degrees of morphisms between moduli spaces.

Connected polynomial degree results to higher genus Gromov--Witten invariants.

Abstract

We study moduli spaces of twisted maps to a smooth pair in arbitrary genus, and give geometric explanations for previously known comparisons between orbifold and logarithmic Gromov--Witten invariants. Namely, we study the space of twisted maps to the universal target and classify its irreducible components in terms of combinatorial/tropical information. We also introduce natural morphisms between these moduli spaces for different rooting parameters and compute their degree on various strata. Combining this with additional hypotheses on the discrete data, we show these degrees are monomial of degree between $0$ and $\max(0,2g-1)$ in the rooting parameter. We discuss the virtual theory of the moduli spaces, and relate our polynomiality results to work of Tseng and You on the higher genus orbifold Gromov--Witten invariants of smooth pairs, recovering their results in genus $1$. We discuss what is needed to deduce arbitrary genus comparison results using the previous sections. We conclude with some geometric examples, starting by re-framing the original genus $1$ example of Maulik in this new formalism.