Blowing Up Finitely Supported Complete Ideals in a Regular Local Ring

TL;DR

This paper investigates the singularities and regularity of the normalization of blow-ups of finitely supported complete ideals in regular local rings, extending previous results to higher dimensions and characterizing the structure of local rings on the normalization.

Contribution

It extends Lipman's and Huneke-Sally's work to higher dimensions, proving regularity of certain local rings on the normalization and characterizing their structure as infinitely near points.

Findings

Normalization of the blow-up is regular if and only if it is obtained by blowing up base points.

Local rings on the normalization that are UFDs are regular.

Such local rings dominate R and are obtained by finite sequences of local quadratic transforms.

Abstract

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.