Thiago Fassarella, Viviana Ferrer, Rodrigo Gondim
We provide a classification of developable cubic hypersurfaces in $\mathbb P^4$. Using the correspondence between forms of degree $3$ on $\mathbb P^4$ and Artinian Gorenstein $\mathbb K$-algebras, given by Macaulay-Matlis duality, we describe the locus in ${\rm GOR}(1,5,5,1)$ corresponding to those algebras which satisfy the Strong Lefschetz property.
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We deal with the Weak Lefschetz property (WLP) for Artinian standard graded Gorenstein algebras of codimension $3.$ We prove that many Gorenstein sequences force the WLP for such algebras. Moreover for every Gorenstein sequence $H$ of codimension 3 we found several Gorenstein Betti sequences compatible with $H$ which again force the WLP. Finally we show that for every Gorenstein Betti sequence the general Artinian standard graded Gorenstein algebra with such Betti sequence has the WLP.
On the Lefschetz locus in Gor(1,n,n,1)
本文研究 Hessian 矩阵消失的三次超曲面，给出两类特殊族的有理参数化与次数，并借助 Macaulay-Matlis 对偶刻画了 Gor(1,n,n,1) 中满足强 Lefschetz 性质的代数轨迹。
We introduce a family of standard bigraded binomial Artinian Gorenstein algebras, whose combinatoric structure characterizes the ones presented by quadrics. These algebras provide, for all socle degree grater than two and in sufficiently large codimension with respect to the socle degree, counter-examples to Migliore-Nagel conjectures, see \cite{MN1} and \cite{MN2}. One of them predicted that Artinian Gorenstein algebras presented by quadratics should satisfy the weak Lefschetz property. We also prove a generalization of a Hessian criterion for the Lefschetz properties given by Watanabe, see \…
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number
本文证明了基座度数为d且Sperner数至多为d+1的artinian Gorenstein代数均满足弱Lefschetz性质，且与余维数无关，并指出该界限在一般情况下是严格的。
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $\ell$ give maximal rank for all the multiplication maps $\times \ell: [A]_i \longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the \emph{non-Lefschetz locus}. For monomial complete …
We introduce a new type of Hessian matrix, that we call Mixed Hessian. The mixed Hessian is used to compute the rank of a multiplication map by a power of a linear form in a standard graded Artinian Gorenstein algebra. In particular we recover the main result of \cite{MW} for identifying Strong Lefschetz elements, generalizing it also for Weak Lefschetz elements. This criterion is also used to give a new proof that Boolean algebras have the Strong Lefschetz Property (SLP). We also construct new examples of Artinian Gorenstein algebras presented by quadrics that does not satisfy the Weak Lefsch…
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Apéry set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I…
In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of…
A theorem of Gordan and Noether via Gorenstein rings
本文以标准Artinian Gorenstein代数与Lefschetz性质为研究视角，给出Gordan-Noether超曲面锥面判定定理的新证法，并证明二次型正则序列生成理想的四维Gorenstein代数具强Lefschetz性质，可应用于光滑三次三维簇雅可比环。
Symmetry, Unimodality, and Lefschetz Properties for Graded Modules
本文研究极小自由分解为三维广义 Koszul 复形的模的弱 Lefschetz 性质，并推广任意维数有限长度模的非 Lefschetz 轨迹结果。
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 m…
An asymptotic description of the Noether-Lefschetz components in toric varieties
本文将 Noether-Lefschetz 分量理论推广到单纯环面簇中的拟光滑超曲面，证明小余维数分量渐近对应于可关联小 k 维子簇的超曲面，并推广了 Green 定理与 artinian Gorenstein 理想概念。
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 $…
Hilbert functions and Jordan type of Perazzo Artinian algebras
本研究聚焦Perazzo超曲面相关的Artinian Gorenstein代数，证明Perazzo三维簇的希尔伯特函数恒单峰且可判定弱Lefschetz性质，同时给出最小希尔伯特函数下三维簇的线性形式乘法若尔当型，还验证了四维簇的希尔伯特函数未必单峰。
Perazzo hypersurfaces and the weak Lefschetz property
本文研究了Perazzo超曲面的Hilbert函数单峰性及弱Lefschetz性质，并证明相关结果不能推广到更高维情形。
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