Geometry & Topology

The moduli space of stable quotients

Alina Marian, Dragos Oprea, and Rahul Pandharipande

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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.

Article information

Geom. Topol., Volume 15, Number 3 (2011), 1651-1706.

Received: 15 March 2011
Revised: 30 May 2011
Accepted: 26 August 2011
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]

Gromov–Witten theory


Marian, Alina; Oprea, Dragos; Pandharipande, Rahul. The moduli space of stable quotients. Geom. Topol. 15 (2011), no. 3, 1651--1706. doi:10.2140/gt.2011.15.1651.

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