  We construct a desingularization of the ``main component'' $\bar{\mathfrak
M}_{1,k}^0(\Bbb{P}^n,d)$ of the moduli space $\bar{\mathfrak
M}_{1,k}(\Bbb{P}^n,d)$ of genus-one stable maps into the complex projective
space $\Bbb{P}^n$. As a bonus, we obtain desingularizations of certain natural
sheaves over $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$. Such desingularizations
are useful for integrating natural cohomology classes on $\bar{\mathfrak
M}_{1,k}^0(\Bbb{P}^n,d)$ 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 $\bar{\mathfrak M}_{1,k}^0(\Bbb{P}^n,d)$ is
obtained by sequentially blowing up $\bar{\mathfrak M}_{1,k}(\Bbb{P}^n,d)$
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.

Algebraic Geometry;Symplectic Geometry