Geometry & Topology
- Geom. Topol.
- Volume 12, Number 1 (2008), 1-95.
A desingularization of the main component of the moduli space of genus-one stable maps into $\mathbb P^n$
We construct a natural smooth compactification of the space of smooth genus-one curves with 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 . In fact, our compactification is obtained from the singular space of stable genus-one maps 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 of . 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.
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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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. https://projecteuclid.org/euclid.gt/1513800013