2006/03/31 by Ravi Vakil, Aleksey Zinger
Mathematics · #Algebraic Geometry and Number Theory #Birational geometry #Compactification (mathematics) #Geometric and Algebraic Topology #Geometric invariant theory #Homotopy and Cohomology in Algebraic Topology #Irreducible component #Moduli space #Morphism #Quintic function #Space (punctuation) #math.AG #math.SG #msc:14D20 #msc:53D99
paper · pdf · doi:10.2140/gt.2008.12.1
published as Geom. Topol. 12 (2008) 1-95 · revised version; 13 figures
arxiv created 2007/02/28 · openalex publication_date 2008/02/08 · arxiv updated 2014/11/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 S M 0;k .P n ; d /. In fact, our compactification is obtained from the singular space of stable genus-one maps S 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 S M 0 1;k .P n ; d/ of S 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.