A Desingularization of the Main Component of the Moduli Space of Genus-One Stable Maps into ${\mathbb{P}}^n$

Abstract

We construct a desingularization of the ``main component'' ${\bar{\mathfrak{M}}}_{1,k}^0({\mathbb{P}}^n,d)$ of the moduli space ${\bar{\mathfrak{M}}}_{1,k}({\mathbb{P}}^n,d)$ of genus-one stable maps into the complex projective space ${\mathbb{P}}^n$. As a bonus, we obtain desingularizations of certain natural sheaves over ${\bar{\mathfrak{M}}}_{1,k}^0({\mathbb{P}}^n,d)$. Such desingularizations are useful for integrating natural cohomology classes on ${\bar{\mathfrak{M}}}_{1,k}^0({\mathbb{P}}^n,d)$ using localization. In turn, these classes can be used to compute the genus-one Gromov-Witten invariants of complete intersections and classical enumerative invariants of projective spaces involving genus-one curves. The desingularization of ${\bar{\mathfrak{M}}}_{1,k}^0({\mathbb{P}}^n,d)$ is obtained by sequentially blowing up ${\bar{\mathfrak{M}}}_{1,k}({\mathbb{P}}^n,d)$ along ``bad'' subvarieties. At the end of the process, we are left with a modification of the main component, which turns out to be nonsingular.

revised version; 13 figures

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