Artinian Gorenstein algebras (AG algebras for short) can be viewed as algebraic analogues of the cohomology rings of smooth projective varieties. The Strong and Weak Lefschetz properties for graded AG algebras take origin from the hard Lefschetz theorem. The properties of an AG quotient $A _F$ of a polynomial ring are related to its Macaulay dual generator $F$, and in particular $A_F$ fails the Strong Lefschetz property if and only if the hessian of $F$ of order $t$ vanishes for some $1\leq t\leq d/2$, where $d=\deg F$ and the usual hessian is obtained for $t=1$. Perazzo polynomials are a large class of polynomials with vanishing hessian so their algebras $A_F$ always fail the SLP. I will report on some recent results concerning the question if the WLP holds for these algebras. Joint work with N. Abdallah, N. Altafi, P. De Poi, L. Fiorindo, A. Iarrobino, P. Macias Marques, R.M. Mir ́o-Roig, L. Nicklasson.

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