The moduli space of stable quotients

TL;DR

This paper introduces a new moduli space of stable quotients for sheaves on curves, providing compactifications, relations in tautological rings, and connections to Gromov-Witten theory and Calabi-Yau geometries.

Contribution

It constructs the moduli space of stable quotients, proves its properties, and relates it to Gromov-Witten invariants and classical geometric spaces.

Findings

Provides a compactification of the moduli of maps from genus 1 curves.

Establishes a virtual class and descendent invariants matching Gromov-Witten theory.

Calculates the conifold invariants aligning with stable maps.

Abstract

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill-Noether constructions. The moduli space of stable quotients is proven to carry a canonical 2-term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the Gromov-Witten theory of the Grassmannian in all genera. Stable quotients can also be used to study Calabi-Yau geometries. The conifold is calculated to agree with stable maps. Several questions about the behavior of stable quotients for arbitrary targets are raised.