Time:  9.30-10h30, Friday, 7 March 2014
Venue: Room 301, Building A5, Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Cau Giay, Ha Noi.

Abstract: The goal of my talk is to establish a close relationship between a priori two unrelated problems:

1. Algebraic Problem: the existence of homogeneous artinian ideals $Isubset k[x_0, cdots, x_n]$ which fail the Weak Lefschetz Property; and
2. Classical Geometric Problem: the existence of (smooth) projective varieties $Xsubset PP^N$ satisfying at least one Laplace equation of order $sgeq 2$. These are two longstanding problems which  lie at the crossroads between Commutative Algebra, Algebraic Geometry, Differential Geometry and Combinatorics.

In the toric case, I will classify some relevant examples  and as byproduct  I will provide counterexamples to Ilardi's conjecture.

Finally, I will classify all smooth Togliatti system of cubics and solve a conjecture stated in my joint work with Mezzetti and Ottaviani.

All I will say is based in joint work with either  E. Mezzetti and G. Ottaviani or M. Michalek.