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A desingularization of the main component of the moduli space of genus-one stable maps into $\mathbb P^n$

Ravi Vakil and Aleksey Zinger

Geometry & Topology 12 (2008) 1–95

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 M̄0,k(n,d). In fact, our compactification is obtained from the singular space of stable genus-one maps M̄1,k(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 M̄1,k0(n,d) of M̄1,k(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.

moduli space of stable maps, genus one, smooth compactification
Mathematical Subject Classification 2000
Primary: 14D20
Secondary: 53D99
Received: 4 March 2007
Revised: 12 October 2007
Accepted: 8 October 2007
Published: 8 February 2008
Proposed: Jim Bryan
Seconded: Gang Tian, Eleny Ionel
Ravi Vakil
Department of Mathematics
Stanford University
Stanford, CA 94305-2125
Aleksey Zinger
Department of Mathematics
SUNY Stony Brook
Stony Brook, NY 11794-3651