Moduli spaces of stable quotients and wall-crossing phenomena

TL;DR

This paper introduces epsilon-stable quotients as a new framework linking stable maps and stable quotients, exploring their wall-crossing behavior and associated Gromov-Witten invariants.

Contribution

It defines epsilon-stable quotients, establishes their relation to stable maps via wall-crossing, and studies the resulting Gromov-Witten invariants.

Findings

Wall-crossing phenomena relate stable maps and stable quotients.

Introduction of epsilon-stable quotients as a new moduli space.

Analysis of Gromov-Witten invariants under wall-crossing.

Abstract

The moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich's stable map compactification and Marian-Oprea-Pandharipande's stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck's Quot scheme. In this paper, we give the notion of `$\epsilon$-stable quotients' for a positive real number $\epsilon$, and show that stable maps and stable quotients are related by the wall-crossing phenomena. We will also discuss Gromov-Witten type invariants associated to $\epsilon$-stable quotients, and investigate them under the wall-crossing.