The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in $\mathbb{P}^n$

TL;DR

This paper investigates the Castelnuovo-Mumford regularity of the Milnor algebra associated with a reduced hypersurface in projective space, revealing bounds related to the Hessian polynomial and demonstrating potential quadratic growth in regularity.

Contribution

It establishes new bounds for the regularity of the Milnor algebra in the presence of positive dimensional singularities and shows that this regularity can grow quadratically with the degree.

Findings

Regularity bounded by (d-2)(n+1) in certain cases

Regularity can grow quadratically with degree d

Hessian polynomial degree relates to regularity bounds

Abstract

For a reduced hypersurface $V(f) \subseteq \mathbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study the regularity of $M(f)$ when $V(f)$ has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by $(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However, this is not always the case, and we prove that in $\mathbb{P}^n$ the regularity of the Milnor algebra can grow quadratically in $d$.