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Journal Articles Advances in Mathematics Year : 2021

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 $d$.

Dates and versions

hal-02337441 , version 1 (29-10-2019)

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Laurent Busé, Alexandru Dimca, Hal Schenck, Gabriel Sticlaru. The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in $\mathbb{P}^n$. Advances in Mathematics, 2021, 392, ⟨10.1016/j.aim.2021.108035⟩. ⟨hal-02337441⟩
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