论文标题
带有消失的黑森和lefschetz属性的多项式
Polynomials with vanishing Hessian and Lefschetz properties
论文作者
论文摘要
目的是研究Perazzo Hypersurfaces $ X = V（f）\ subseteq \ Mathbb {p}（k^5）$，由$ f（x_0，x_1，x_1，x_2，u，u，u，v）= p_0（u，v） $ p_0，p_1，p_2 $是代数依赖的，但是在$ u，v $和$ g $中的$ d-1 $的独立形式是$ u，v $ of级$ d $的形式。这些Hypersurfaces是$ \ Mathbb {p}^4 $中所有可能的Hypersuface的“构建块”，并消失了Hessian。对于关联的Artinian Gorenstein $ k $ -k $ -algebras $ a_f $的相关的Artinian Gorenstein $ k $ a_f $的最小和最大的希尔伯特矢量：在最小情况下，它们满足了弱的lefschetz物业，但在最大情况下，它们不满意。此外，我们将所有Perazzo $ 3 $ folds归类为最小$ h $ - vector。我们还总结了有关Hypersurfaces的基本知识和已经知道的结果，其中消失的Hessian及其几何形状在低维度以及Artinian Gorenstein $ k $ -k $ -algebras。
The aim is to study Perazzo hypersurfaces $X=V(F)\subseteq\mathbb{P}(K^5)$, defined by $F(x_0,x_1,x_2,u,v) = p_0(u,v)x_0+p_1(u,v)x_1+p_2(u,v)x_2+g(u,v)$, where $p_0,p_1,p_2$ are algebraically dependent, but linearly independent forms of degree $d-1$ in $u,v$, and $g$ is a form in $u,v$ of degree $d$. These hypersurfaces are the "building blocks" for all possible hypersuface in $\mathbb{P}^4$ with vanishing Hessian. A minimal and a maximal Hilbert vector is found for the associated Artinian Gorenstein $K$-algebras $A_F$: in the minimal case they satisfy the Weak Lefschetz property, but in the maximal case they don't. Furthermore, we classify all Perazzo $3$-folds with minimal $h$-vector. We also summarise basic knowledge and already known results about hypersurfaces with vanishing Hessian and their geometry in low dimension, and also about Artinian Gorenstein $K$-algebras.
