تقرير
A theorem of Gordan and Noether via Gorenstein rings
| العنوان: | A theorem of Gordan and Noether via Gorenstein rings |
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| المؤلفون: | Bricalli, Davide, Favale, Filippo F., Pirola, Gian Pietro |
| المصدر: | Selecta Mathematica (2023) |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Commutative Algebra, Primary: 14J70, Secondary: 14J30, 13E10, 14A25, 14A05 |
| الوصف: | Gordan and Noether proved in their fundamental theorem that an hypersurface $X=V(F)\subseteq \mathbb{P}^n$ with $n\leq 3$ is a cone if and only if $F$ has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if $n\geq 4$, by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein $\mathbb{K}$-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra $R=\mathbb{K}[x_0,\dots,x_4]/J$ with $J$ generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds. Comment: 21 pages |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1007/s00029-023-00882-7 |
| URL الوصول: | http://arxiv.org/abs/2201.07550 |
| رقم الأكسشن: | edsarx.2201.07550 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1007/s00029-023-00882-7 |
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