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

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 $d$.

v.4: Final version, to appear in Advances in Mathematics