Gromov-Witten/Hurwitz wall-crossing

TL;DR

This paper introduces a new stability condition for maps from nodal curves to product varieties, establishes wall-crossing formulas for invariants, and proves the crepant resolution conjecture for certain toric del Pezzo surfaces.

Contribution

It develops the concept of $psilon$-admissibility for maps, derives wall-crossing formulas, and connects these to the crepant resolution conjecture in algebraic geometry.

Findings

Established wall-crossing formulas relating invariants for different $psilon$ values.

Proved the crepant resolution conjecture for 3-point genus-0 invariants on toric del Pezzo surfaces.

Connected relative Gromov-Witten theory with Ruan's extended crepant resolution conjecture.

Abstract

For a target variety $X$ and a nodal curve $C$, we introduce a one-parameter stability condition, termed $\epsilon$-admissibility, for maps from nodal curves to $X\times C$. If $X$ is a point, $\epsilon$-admissibility interpolates between moduli spaces of stable maps to $C$ relative to some fixed points and moduli spaces of admissible covers with arbitrary ramifications over the same fixed points and simple ramifications elsewhere on $C$. Using Zhou's entangled tails, we prove wall-crossing formulas relating invariants for different values of $\epsilon$. If $X$ is a surface, we use this wall-crossing in conjunction with author's quasimap wall-crossing to show that the relative Pandharipande-Thomas/Gromov-Witten correspondence of $X\times C$ and Ruan's extended crepant resolution conjecture of the pair $X^{[n]}$ and $[X^{(n)}]$ are equivalent up to explicit wall-crossings. We thereby prove the crepant resolution conjecture for 3-point genus-0 invariants in all classes, if $X$ is a toric del Pezzo surface.