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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 M0,k(Pn,d).
In fact, our compactification is obtained from the singular space of stable
genus-one maps M1,k(Pn,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 M1,k0(Pn,d) of M1,k(Pn,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.
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