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

Abstract

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 ${\bar{\mathfrak{M}}}_{0,k}\left({\mathbb{P}}^n,d\right)$. In fact, our compactification is obtained from the singular space of stable genus-one maps ${\bar{\mathfrak{M}}}_{1,k}\left({\mathbb{P}}^n,d\right)$ 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 ${\bar{\mathfrak{M}}}_{1,k}^0\left({\mathbb{P}}^n,d\right)$ of ${\bar{\mathfrak{M}}}_{1,k}\left({\mathbb{P}}^n,d\right)$. 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.