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 $d$.