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A natural smooth compactification of the space of elliptic curves in projective space

Authors: Ravi Vakil and Aleksey Zinger
Journal: Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 53-59
MSC (2000): Primary 14D20; Secondary 53D99
Published electronically: June 11, 2007
MathSciNet review: 2320682
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Abstract: The space of smooth genus-0 curves in projective space has a natural smooth compactification: the moduli space of stable maps, which may be seen as the generalization of the classical space of complete conics. In arbitrary genus, no such natural smooth model is expected, as the space satisfies ``Murphy's Law''. In genus $ 1$, however, the situation remains beautiful. We give a natural smooth compactification of the space of elliptic curves in projective space, and describe some of its properties. This space is a blowup of the space of stable maps. It can be interpreted as a result of blowing up the most singular locus first, then the next most singular, and so on, but with a twist--these loci are often entire components of the moduli space. We give a number of applications in enumerative geometry and Gromov-Witten theory. For example, this space is used by the second author to prove physicists' predictions for genus-1 Gromov-Witten invariants of a quintic threefold. The proof that this construction indeed gives a desingularization will appear in a subsequent paper.

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Additional Information

Ravi Vakil
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125

Aleksey Zinger
Affiliation: Department of Mathematics, SUNY Stony Brook, Stony Brook, New York 11794-3651

Received by editor(s): July 13, 2006
Published electronically: June 11, 2007
Additional Notes: The first author was partially supported by PECASE/CAREER grant DMS-0238532. The second author was partially supported by a Sloan Fellowship and NSF Grant DMS-0604874
Communicated by: János Kollár
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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