Perazzo hypersurfaces and the Lefschetz properties

Abstract

[en] The main goal of this writing is to introduce basic concepts of algebraic geometry and commutative algebra to be able to study the relationship between Perazzo hypersurfaces and the Lefschetz properties. We will introduce graded algebras and how to construct one of them from a hypersurface to further check if they satisfy any of the properties. In this paperwork, we have found an upper bound and a low enough value of the Hilbert vector for Perazzo hypersurfaces. The notable result we have obtained is that the Weak Lefschetz property is failed to obtain when the h-vector is maximal, and conversely, it is always obtained when the h-vector is on that low value.