Stable maps and stable quotients

TL;DR

This paper explores the relationship between two compactifications of the moduli space of maps to Grassmannians, constructing a unifying space and relating their enumerative invariants through virtual fundamental classes.

Contribution

It introduces a new moduli space dominating both stable maps and stable quotients, and proves the equivalence of their enumerative invariants.

Findings

Constructed a moduli space dominating both stable maps and stable quotients.

Established a relation between the virtual fundamental classes of the three moduli spaces.

Provided a new proof that stable quotient invariants coincide with Gromov--Witten invariants.

Abstract

We analyze the relationship between two compactifications of the moduli space of maps from curves to a Grassmannian: the Kontsevich moduli space of stable maps and the Marian--Oprea--Pandharipande moduli space of stable quotients. We construct a moduli space which dominates both the moduli space of stable maps to a Grassmannian and the moduli space of stable quotients, and equip our moduli space with a virtual fundamental class. We relate the virtual fundamental classes of all three moduli spaces using the virtual push-forward formula. This gives a new proof of a theorem of Marian-Oprea-Pandharipande: that enumerative invariants defined as intersection numbers in the stable quotient moduli space coincide with Gromov--Witten invariants.