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Stable maps and stable quotients

Part of: Families, fibrations Projective and enumerative geometry

Published online by Cambridge University Press:  17 July 2014

Cristina Manolache
Cristina Manolache*
Affiliation:
Imperial College London, 180 Queen’s Gate, SW7 2AZ London, UK email [email protected]
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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.

Keywords

Gromov–Witten invariantsmoduli spacesintersection theory

MSC classification

Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14D23: Stacks and moduli problems
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 9 , September 2014 , pp. 1457 - 1481
Copyright
© The Author 2014 

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