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.
Keywords
moduli space of stable maps, genus one, smooth
compactification