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The moduli space of stable quotients

Alina Marian, Dragos Oprea and Rahul Pandharipande

Geometry & Topology 15 (2011) 1651–1706

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.

Dedicated to William Fulton on the occasion of his 70th birthday

Gromov–Witten theory
Mathematical Subject Classification 2010
Primary: 14N35
Secondary: 14C17
Received: 15 March 2011
Revised: 30 May 2011
Accepted: 26 August 2011
Published: 1 October 2011
Proposed: Jim Bryan
Seconded: Richard Thomas, Frances Kirwan
Alina Marian
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
851 S Morgan Street
Chicago IL 60607
Dragos Oprea
Department of Mathematics
University of California San Diego
9500 Gilman Drive #0112
La Jolla CA 92093-0112
Rahul Pandharipande
Department of Mathematics
Princeton University
406 Fine Hall
Princeton NJ 08544