Higher Nash blowups of the A3-singularity

Abstract

We show that the n-th Nash blowup of the toric surface singularity of type A3 is singular for any . It has been known that the normalization of the n-th Nash blowup of an affine normal toric variety is the toric variety associated to the Gröbner fan of a certain ideal. In our case, we prove that the Gröbner fan contains a certain non-regular cone for any . Thus we conclude that the normalizations are singular, and so are the Nash blowups.