Blowing up finitely supported complete ideals in a regular local ring

Abstract

Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m). We consider singularities of the projective models ProjR[It] and ProjR[It]‾ over SpecR, where R[It]‾ denotes the integral closure of the Rees algebra R[It]. A theorem of Lipman implies that the ideal I has a unique factorization as a ⁎-product of special ⁎-simple complete ideals with possibly negative exponents for some of the factors. If ProjR[It]‾ is regular, we prove that ProjR[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 ProjR[It]‾ that is a unique factorization domain is regular. Moreover, if dim⁡S≥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.