  Let I be a finitely supported complete m-primary ideal of a regular local
ring (R, m). We consider singularities of the normalization of the blow-up Proj
R[It] of I. A theorem of Lipman implies that the ideal I has a unique
factorization as a star-product of special star-simple complete ideals with
possibly negative exponents for some of the factors. If the normalization of
the projective model Proj R[It] is regular, we prove that it is the regular
model obtained by blowing up the finite set of base points of I. Extending work
of Lipman and Huneke-Sally in dimension 2, we prove that every local ring S on
the normalization of Proj R[It] that is a unique factorization domain is
regular. Moreover, if dim S is at least 2 and S dominates R, then S is an
infinitely near point to R, that is, S is obtained from R by a finite sequence
of local quadratic transforms.
