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.

中文翻译：

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Pn 中简化超曲面的 Hessian 多项式和 Jacobian 理想

对于减少的超曲面 $伏(F)\subseteq {\mathbb{磷}}^n$度d的代数米尔诺的卡斯德尔诺-芒福德规律性$米(F)$ 很好理解当 $伏(F)$ 是平滑的，以及当 $伏(F)$有孤立的奇点。我们研究的规律$米(F)$ 什么时候 $伏(F)$具有正维奇异轨迹。在某些情况下，我们证明正则性有界$(d-2)(n+1)$，这是f的 Hessian 多项式的次数。然而，情况并非总是如此，我们证明在${\mathbb{磷}}^n$Milnor 代数的正则性可以在d 中二次增长。