论文标题
Gordan定理和Noether通过Gorenstein Rings
A theorem of Gordan and Noether via Gorenstein rings
论文作者
论文摘要
Gordan和Noether在其基本定理中证明了Hypersurface $ X = V（f）\ subseteq \ Mathbb {p}^n $，with $ n \ leq 3 $是锥体，并且仅当$ f $消失了Hessian（即Hessian Matrix的决定性）时。他们还表明，如果$ n \ geq 4 $，通过给出一些反例，则该声明是错误的。自从他们的证据以来，文献中已经提出了其他一些人。在本文中，我们通过使用不同的观点给出了一个新的，涉及对标准的Artinian Gorenstein $ \ Mathbb {K} $ - 代数和Lefschetz属性进行研究。作为我们设置的进一步应用，我们证明了标准的Artinian Gorenstein代数$ r = \ Mathbb {k} [x_0，\ dots，x_4]/j $，$ j $由常规序列产生的$ j $具有强大的lefschetz属性。特别是，这适用于与平滑立方三倍相关的雅各布环。
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.
