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A
desingularization of the main component of the moduli space
of genus-one stable maps into $\mathbb P^n$
Ravi Vakil and Aleksey Zinger |
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Geometry & Topology 12 (2008) 1–95
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DOI: 10.2140/gt.2008.12.1
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Abstract |
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We construct a natural smooth compactification of the space of smooth genus-one curves with distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps . In fact, our compactification is obtained from the singular space of stable genus-one maps through a natural sequence of blowups along “bad” subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space. As a bonus, we obtain desingularizations of certain natural sheaves over the “main” irreducible component of . A number of applications of these desingularizations in enumerative geometry and Gromov–Witten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus-one Gromov–Witten invariants of a quintic threefold. |
Keywords
moduli space of stable maps, genus one, smooth
compactification
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Mathematical Subject Classification 2000
Primary: 14D20
Secondary: 53D99
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References |
Publication
Received: 4 March 2007
Revised: 12 October 2007
Accepted: 8 October 2007
Published: 8 February 2008
Proposed: Jim Bryan
Seconded: Gang Tian, Eleny Ionel
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Authors |
| Ravi Vakil | |
| Department of Mathematics Stanford University Stanford, CA 94305-2125 USA |
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| http://math.stanford.edu/~vakil | |
| Aleksey Zinger | |
| Department of Mathematics SUNY Stony Brook Stony Brook, NY 11794-3651 USA |
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| http://www.math.sunysb.edu/~azinger | |
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