A Nobile-like theorem for jet schemes of hypersurfaces

TL;DR

This paper establishes a new Nobile-like theorem for jet schemes of singular hypersurfaces, demonstrating that a specific jet-related module blowup is not an isomorphism, extending results across arbitrary fields.

Contribution

It introduces a novel Nobile-like theorem for jet schemes of hypersurfaces using explicit Jacobian matrix presentations, applicable in any characteristic.

Findings

The blowup of a jet-related module is not an isomorphism for singular hypersurfaces.

The result extends to fields of arbitrary characteristic.

Uses higher-order Jacobian matrices and jet-related matrices for proofs.

Abstract

We prove that, for the jet scheme of a singular hypersurface, the blowup of a certain jet-related module is not an isomorphism. In conjunction with recent developments in the theory of Nash blowups, our result holds over fields of arbitrary characteristic. Our approach is based on explicit presentations given by a higher-order Jacobian matrix combined with a certain jet-related matrix.