A desingularization of the main component of the moduli space of genus-one stable maps into ℙn

We construct a natural smooth compactification of the space of smooth genus-one curves with $k$ 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 ${\bar{M}}_{0, k} (ℙ^{n}, d)$ . In fact, our compactification is obtained from the singular space of stable genus-one maps ${\bar{M}}_{1, k} (ℙ^{n}, d)$ 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 ${\bar{M}}_{1, k}^{0} (ℙ^{n}, d)$ of ${\bar{M}}_{1, k} (ℙ^{n}, d)$ . 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

Mathematical Subject Classification 2000

Primary: 14D20

Secondary: 53D99

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

Authors

Ravi Vakil
Department of Mathematics Stanford University Stanford, CA 94305-2125 USA
http://math.stanford.edu/~vakil
Aleksey Zinger
Department of Mathematics SUNY Stony Brook Stony Brook, NY 11794-3651 USA
http://www.math.sunysb.edu/~azinger

Volume 12, issue 1 (2008)

Ravi Vakil and Aleksey Zinger

Abstract

Keywords

Mathematical Subject Classification 2000

References

Publication

Authors