# The Hessian polynomial and the Jacobian ideal of a reduced surface in $\mathbb{P}^3$

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
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 $\mathbb{P}^3$ the regularity of the Milnor algebra can grow quadratically in $d$.
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Document type :
Preprints, Working Papers, ...
Domain :

https://hal.inria.fr/hal-02337441
Contributor : Laurent Busé <>
Submitted on : Tuesday, October 29, 2019 - 2:28:33 PM
Last modification on : Thursday, January 21, 2021 - 1:34:13 PM

### Identifiers

• HAL Id : hal-02337441, version 1
• ARXIV : 1910.09195

### Citation

Laurent Busé, Alexandru Dimca, Hal Schenck, Gabriel Sticlaru. The Hessian polynomial and the Jacobian ideal of a reduced surface in $\mathbb{P}^3$. 2020. ⟨hal-02337441⟩

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