Type: Article

On the zero locus of ideals defining the Nash blowup of toric surfaces

Martínez, Enrique Chávez; Duarte, Daniel;

Abstract:

The Nash blowing up of an algebraic variety X is the process of replacing each singular point of X by the limit of tangent spaces at non-singular points. This modification of X is isomorphic to the blowup of an ideal sheaf on X, say J, whose zero locus V(J) contains the singular locus Sing(X) of X. In particular, if X is a complete intersection, we have Sing(X)=V(J). In this paper, the authors study when we have this coincidence for non-complete intersection varieties. They are able to prove that if X is a non-complete intersection toric surface with dimSing(X)=1, then there is a choice of an ideal J defining the Nash blowup of X such that Sing(X)=V(J), and if dimSing(X)=0, there does not exist such an ideal.