The Ubiquity of the weak Lefschetz property
Speaker: Rosa M. Miró-Roig (University of Barcelona)

Time: 9h00, Wednesday, September 6, 2017
Room 6, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi
I will show once more the ubiquity of the Weak Lefschetz Property (WLP). First, I will establish a close relationship between  a priori two unrelated problems: (1)  the existence of Togliatti systems (i.e. homogeneous artinian ideals $Isubset k[x_0, cdots , x_n]$  generated by forms of degree $d$ which fail the WLP in degree $d-1$); and (2) the existence of (smooth) projective varieties $Xsubset PP^N$ satisfying at least one Laplace equation of order $d-1geq 2$. These are two longstanding problems which lie at the crossroads between Commutative Algebra, Algebraic Geometry, Differential Geometry, and Combinatorics.
In the monomial case, I will classify some relevant examples and as byproduct I will provide counterexamples to Ilardi's conjecture. Then, I will classify all smooth Togliatti system of quadrics and cubics and solve a conjecture stated in my joint work with Mezzetti and Ottaviani.
For $dge 4$, the picture becomes soon much more involved and a complete classification is out of reach. Nevertheless, we have been able to establish minimal and maximal bounds, depending on $n$ and $dge 2$, for the number of generators of  Togliatti systems and to classify the systems reaching the minimal bound, or close to reach it. I will also investigate if all values comprised between the minimal and the maximal one are reached as the number of generators of a minimal smooth Togliatti system.
Finally, I will relate Galois coverings with cyclic group $Z /dZ$ to monomial Togliatti systems of forms of degree $d$.