Propriedade fraca de Lefschetz e a classificação de Sistemas de Togliatti minimais monomiais

Abstract

Classifying smooth varieties that satisfy at least one Laplace equation is an ancient problem in algebraic and differential geometry, as can be seen in [28] and [27], where E. Togliatti provided one of the earliest contributions to this problem. He proved that there exists one and only one example of a rational surface in P 5 parametrized by cubics and satisfying a Laplace equation of order 2. In [20], E. Mezzetti, G. Ottaviani and R. M. Miró-Roig proved that there is a relationship between the existence of projective varieties X ⊂ P N satisfying at least one aplace equation of order s ≥ 2 and the existence of homogeneous Artinian ideals I ⊂ R = κ[x0, · · · , xn] generated by forms of degree d that fail the weak Lefschetz property in degree d − 1. They showed that an Artinian ideal I ⊂ R generated by r forms of degree d, where r ≤ d+n−1 n−1 ,fails the weak Lefschetz property in degree d−1 if, and only if, the projection of the Veronese variety V (n, d) by the linear system |I −1d|, denoted by XI −1 d , has osculatory defect of order d − 1. In this case, I is called a Togliatti system. Although the problem of classifying all projective varieties that have osculatory defect, and consequently, all Togliatti systems, seems to be out of reach at the moment, in this work, we will focus our efforts on the study of the monomial case, since in this case, the associated variety XI −1 d is toric, and various combinatorial tools can be used for the study of Togliatti systems.