Geometry & Topology

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

Ravi Vakil and Aleksey Zinger

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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.

Article information

Geom. Topol., Volume 12, Number 1 (2008), 1-95.

Received: 4 March 2007
Revised: 12 October 2007
Accepted: 8 October 2007
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
Secondary: 53D99: None of the above, but in this section

moduli space of stable maps genus one smooth compactification


Vakil, Ravi; Zinger, Aleksey. A desingularization of the main component of the moduli space of genus-one stable maps into $\mathbb P^n$. Geom. Topol. 12 (2008), no. 1, 1--95. doi:10.2140/gt.2008.12.1.

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