Deforming spaces of m-jets of hypersurfaces singularities

TL;DR

This paper studies how certain deformations of hypersurface singularities induce corresponding deformations of their jet spaces, especially when the deformations admit simultaneous resolutions, with applications to nondegenerate cases.

Contribution

It establishes conditions under which embedded deformations of hypersurfaces induce deformations of their jet schemes, extending understanding of singularity deformations and jet space behavior.

Findings

Embedded deformations with simultaneous resolutions induce jet space deformations.

Nondegenerate Newton boundary deformations admit such induced jet deformations.

Provides examples involving hypersurfaces with isolated singularities.

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, and $V$ a hypersurface defined by an irreducible polynomial $f$ with coefficients in $\mathbb{K}$ . In this article we prove that an Embedded Deformation of $V$ which admits a Simultaneous Embedded Resolution induces, under certain mild conditions, a deformation of the reduced scheme associated to the space of $m$-jets $V_m$, $m\geq 0$. An example of an Embedded Deformation of $V$ which admits a Simultaneous Embedded Resolution is a $\Gamma(f)$-deformation of $V$, where $V$ has at most one isolated singularity, and $f$ is non degenerate with respect to the Newton Boundary $\Gamma(f)$.