Luca Fiorindo: Perazzo algebras and the weak Lefschetz property

Abstract.

A Perazzo polynomial is a form F of type F=p_0 x_0+p_1 x_1+ p_2 x_2+g where p_0,p_1,p_2, and g are forms in the two variables u,v. These polynomials always have vanishing Hessian. They were first studied by Gordan and Noether in 1876, and then by Perazzo in 1900 in a geometric way. In the algebraic picture, we study "Perazzo algebras": a Perazzo algebra A_F is an artinian Gorenstein algebra with F as dual generator. The property of F having vanishing Hessian is translated to A_F as the failure of the strong Lefschetz property. This talk will present a study of the Perazzo algebras using both algebraic and geometric tools. The principal question is: "Does A_F fail the weak Lefschetz property or not?" . This is a joint work with N. Abdallah, N. Altafi, P. de Poi, A. Iarrobino, P. Macias Marques, E. Mezzetti, R. M. Miró-Roig, and L. Nicklasson.