Higher Nash Blowups of $A_3$-Singularity

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 $n$.