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 | References | Similar Articles | Additional Information

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 , 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.

**[BCOV]**M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa,*Holomorphic anomalies in topological field theories*, Nuclear Phys. B**405**(1993), no. 2-3, 279–304. MR**1240687**, 10.1016/0550-3213(93)90548-4**[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 genus-one Gromov-Witten invariants of a quintic threefold*, preprint 2004, math.AG/0406105.**[LZ2]**J. Li and A. Zinger,*On the genus-one Gromov-Witten invariants of complete intersections*, preprint 2005, math.AG/0507104.**[P]**Rahul Pandharipande,*Intersections of 𝐐-divisors on Kontsevich’s moduli space \overline𝐌_{0,𝐧}(𝐏^{𝐫},𝐝) and enumerative geometry*, Trans. Amer. Math. Soc.**351**(1999), no. 4, 1481–1505. MR**1407707**, 10.1090/S0002-9947-99-01909-1**[V1]**Ravi Vakil,*The enumerative geometry of rational and elliptic curves in projective space*, J. Reine Angew. Math.**529**(2000), 101–153. MR**1799935**, 10.1515/crll.2000.094**[V2]**Ravi Vakil,*Murphy’s law in algebraic geometry: badly-behaved deformation spaces*, Invent. Math.**164**(2006), no. 3, 569–590. MR**2227692**, 10.1007/s00222-005-0481-9**[VZ]**R. Vakil and A. Zinger,*A desingularization of the main component of the moduli space of genus-one stable maps to projective space*, preprint 2006, math.AG/0603353v1.**[Z1]**A. Zinger,*Reduced genus-one Gromov-Witten invariants*, preprint 2005, math.SG/0507103.**[Z2]**A. Zinger,*Intersections of tautological classes on blowups of moduli spaces of genus-one curves*, preprint 2006, math.AG/0603357.**[Z3]**A. Zinger,*The reduced genus-one Gromov-Witten invariants of Calabi-Yau hypersurfaces*, preprint 2007, math/0705.2397.

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Additional Information

**Ravi Vakil**

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

Email:
vakil@math.stanford.edu

**Aleksey Zinger**

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

Email:
azinger@math.sunysb.edu

DOI:
https://doi.org/10.1090/S1079-6762-07-00174-6

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.