Higher Nash Blowups of $A_3$-Singularity

TL;DR

This paper demonstrates that the higher Nash blowups of the $A_3$-toric surface singularity remain singular for all positive n, revealing complex geometric structures through Gr"obner fan analysis.

Contribution

It proves the singularity of all higher Nash blowups of the $A_3$-singularity and analyzes their Gr"obner fans and initial ideals.

Findings

Higher Nash blowups of $A_3$ are always singular.

The Gr"obner fan contains non-regular cones.

Explicit minimal generators of initial ideals are determined.

Abstract

We show that the $ n $-th Nash blowup of the toric surface singularity of type $ A_3 $ is singular for any $ n > 0 $. It was known that the normalization of the $ n $-th Nash blowup of a toric variety is also a toric variety associated to the Gr\"{o}bner fan of a certain ideal $ J_n $. In our case, we prove that the Gr\"{o}bner fan contains a non-regular cone. We determine minimal generators of the initial ideal of $ J_n $ with respect to a certain monomial ordering, and show that the reduced Gr\"{o}bner basis of $ J_n $ has polynomials of certain forms for each $ n $.