Computational aspects of the higher Nash blowup

TL;DR

This paper introduces a higher-order Jacobian matrix to explicitly compute the higher Nash blowup of hypersurfaces, generalizing existing methods and providing new insights into the structure of these modifications.

Contribution

It defines a higher-order Jacobian matrix and uses it to explicitly compute the higher Nash blowup of hypersurfaces, extending previous techniques and results.

Findings

Explicit computation method for higher Nash blowup of hypersurfaces

Description of the ideal defining the blowup

A higher-order Nobile's theorem for normal hypersurfaces

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 this note we will define a higher-order Jacobian matrix that will allow us to make explicit computations concerning the higher Nash blowup of hypersurfaces. Firstly, we will generalize a known method to compute the fiber of this modification. Secondly, we will give an explicit description of the ideal whose blowup gives the higher Nash blowup. As a consequence, we will deduce a higher-order version of Nobile's theorem for normal hypersurfaces.