论文标题
Perazzo 3倍和弱Lefschetz物业
Perazzo 3-folds and the weak Lefschetz property
论文作者
论文摘要
我们在$ \ mathbb p^4 $中处理Perazzo 3倍，即HyperSurfaces $ x = V（f） $ f（x_0，x_1，x_2，u，u，v）= p_0（u，v）x_0+p_1（u，v）x_1+p_1+p_2（u，v）x_2+g（u，v）$，其中$ p_0，p_0，p_1，p_1，p_2 $是elgebrailly依赖的，但$ d-1 $ $ d-1 $ d-1 $ d-1学位$ D $。 Perazzo 3倍消失的Hessian，因此，相关分级的Artinian Gorenstein代数$ A_F $失败了强大的Lefschetz物业。在本文中，我们确定了$ a_f $的最大和最小Hilbert功能，我们证明，如果$ a_f $具有最大的Hilbert功能，则会使弱Lefschetz属性失败，而当它具有最低Hilbert功能时，它会满足弱Lefschetz属性。此外，我们将所有perazzo分为$ \ mathbb p^4 $中的所有perazzo，以使$ a_f $具有最低hilbert功能。
We deal with Perazzo 3-folds in $\mathbb P^4$, i.e. hypersurfaces $X=V(f)\subset \mathbb P^4$ of degree $d$ defined by a homogeneous polynomial $f(x_0,x_1,x_2,u,v)=p_0(u,v)x_0+p_1(u,v)x_1+p_2(u,v)x_2+g(u,v)$, 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 $d$. Perazzo 3-folds have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra $A_f$ fails the strong Lefschetz property. In this paper, we determine the maximum and minimum Hilbert function of $A_f$ and we prove that if $A_f$ has maximal Hilbert function it fails the weak Lefschetz property, while it satisfies the weak Lefschetz property when it has minimum Hilbert function. In addition, we classify all Perazzo 3-folds in $\mathbb P^4$ such that $A_f$ has minimum Hilbert function.
