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), 5359
MSC (2000):
Primary 14D20; Secondary 53D99
Published electronically:
June 11, 2007
MathSciNet review:
2320682
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Abstract: The space of smooth genus0 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 , 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 twistthese loci are often entire components of the moduli space. We give a number of applications in enumerative geometry and GromovWitten theory. For example, this space is used by the second author to prove physicists' predictions for genus1 GromovWitten invariants of a quintic threefold. The proof that this construction indeed gives a desingularization will appear in a subsequent paper.
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R. Vakil and A. Zinger, A desingularization of the main component of the moduli space of genusone stable maps to projective space, preprint 2006, math.AG/0603353v1.
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A. Zinger, Reduced genusone GromovWitten invariants, preprint 2005, math.SG/0507103.
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A. Zinger, Intersections of tautological classes on blowups of moduli spaces of genusone curves, preprint 2006, math.AG/0603357.
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A. Zinger, The reduced genusone GromovWitten invariants of CalabiYau hypersurfaces, preprint 2007, math/0705.2397.
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 M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993), 279304. MR 1240687 (94j:81254)
 [F]
 C. Fontanari, Towards the cohomology of moduli spaces of higher genus stable maps, preprint 2006, math.AG/0611754.
 [LZ1]
 J. Li and A. Zinger, On the genusone GromovWitten invariants of a quintic threefold, preprint 2004, math.AG/0406105.
 [LZ2]
 J. Li and A. Zinger, On the genusone GromovWitten invariants of complete intersections, preprint 2005, math.AG/0507104.
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 R. Pandharipande, Intersections of divisors on Kontsevich's moduli space and enumerative geometry, Trans. Amer. Math. Soc. 351 (1999), no. 4, 14811505. MR 1407707 (99f:14068)
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 R. Vakil, The enumerative geometry of rational and elliptic curves in projective space, J. Reine Angew. Math. (Crelle's Journal) 529 (2000), 101153. MR 1799935 (2001j:14072)
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 R. Vakil, Murphy's Law in algebraic geometry: Badlybehaved moduli deformation spaces, Invent. Math. 164 (2006), 569590. MR 2227692 (2007a:14008)
 [VZ]
 R. Vakil and A. Zinger, A desingularization of the main component of the moduli space of genusone stable maps to projective space, preprint 2006, math.AG/0603353v1.
 [Z1]
 A. Zinger, Reduced genusone GromovWitten invariants, preprint 2005, math.SG/0507103.
 [Z2]
 A. Zinger, Intersections of tautological classes on blowups of moduli spaces of genusone curves, preprint 2006, math.AG/0603357.
 [Z3]
 A. Zinger, The reduced genusone GromovWitten invariants of CalabiYau hypersurfaces, preprint 2007, math/0705.2397.
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Additional Information
Ravi Vakil
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 943052125
Email:
vakil@math.stanford.edu
Aleksey Zinger
Affiliation:
Department of Mathematics, SUNY Stony Brook, Stony Brook, New York 117943651
Email:
azinger@math.sunysb.edu
DOI:
http://dx.doi.org/10.1090/S1079676207001746
PII:
S 10796762(07)001746
Received by editor(s):
July 13, 2006
Published electronically:
June 11, 2007
Additional Notes:
The first author was partially supported by PECASE/CAREER grant DMS0238532. The second author was partially supported by a Sloan Fellowship and NSF Grant DMS0604874
Communicated by:
János Kollár
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
