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 Proj R [ I t ] and Proj R [ I t ] ‾ over Spec  R , where R [ I t ] ‾ denotes the integral closure of the Rees algebra R [ I t ] . 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 Proj R [ I t ] ‾ is regular, we prove that Proj R [ I t ] ‾ 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 Proj R [ I t ] ‾ 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.