Higher Nash blowup on normal toric varieties

TL;DR

This paper provides a combinatorial description of the higher Nash blowup for normal toric varieties using Gr"obner fans, and proves it is an isomorphism only for non-singular cases.

Contribution

It introduces a combinatorial framework for higher Nash blowups on normal toric varieties and establishes an analogue of Nobile's theorem in this setting.

Findings

Higher Nash blowup is described via Gr"obner fans.

The higher Nash blowup is an isomorphism iff the variety is non-singular.

The result extends Nobile's theorem to higher Nash blowups.

Abstract

The higher Nash blowup of an algebraic variety replaces singular points with limits of certain spaces carrying higher order data associated to the variety at non-singular points. In the case of normal toric varieties we give a combinatorial description of the higher Nash blowup in terms of a Gr\"obner fan. This description will allow us to prove the analogue of Nobile's theorem on the usual Nash blowup in this context. More precisely, we prove that for a normal toric variety, the higher Nash blowup is an isomorphism if and only if the variety is non-singular.