A natural smooth compactification of the space of elliptic curves in projective space
Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 53-59
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.
@article{10_1090_S1079_6762_07_00174_6,
author = {Vakil, Ravi and Zinger, Aleksey},
title = {A natural smooth compactification of the space of elliptic curves in projective space},
journal = {Electronic research announcements of the American Mathematical Society},
pages = {53--59},
year = {2007},
volume = {13},
doi = {10.1090/S1079-6762-07-00174-6},
url = {http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-07-00174-6/}
}
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Vakil, Ravi; Zinger, Aleksey. A natural smooth compactification of the space of elliptic curves in projective space. Electronic research announcements of the American Mathematical Society, Tome 13 (2007), pp. 53-59. doi: 10.1090/S1079-6762-07-00174-6
[1] , , , Holomorphic anomalies in topological field theories Nuclear Phys. B 1993 279 304
[2] Intersections of 𝐐-divisors on Kontsevich’s moduli space \overline𝐌_{0,𝐧}(𝐏^{𝐫},𝐝) and enumerative geometry Trans. Amer. Math. Soc. 1999 1481 1505
[3] The enumerative geometry of rational and elliptic curves in projective space J. Reine Angew. Math. 2000 101 153
[4] Murphy’s law in algebraic geometry: badly-behaved deformation spaces Invent. Math. 2006 569 590
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