The Lefschetz question for ideals generated by powers of linear forms in few variables

Abstract

The Lefschetz question asks if multiplication by a power of a general linear form, L, on a graded algebra has maximal rank (in every degree). We consider a quotient by an ideal that is generated by powers of linear forms. Then the Lefschetz question is, for example, related to the problem whether a set of fat points imposes the expected number of conditions on a linear system of hypersurfaces of fixed degree. Our starting point is a result that relates Lefschetz properties in different rings. It suggests to use induction on the number of variables, n. If n=3, then it is known that multiplication by L always has maximal rank. We show that the same is true for multiplication by L2 if all linear forms are general. Furthermore, we give a complete description of when multiplication by L3 has maximal rank (and its failure when it does not). As a consequence, for such ideals that contain a quadratic or cubic generator, we establish results on the so-called strong Lefschetz property for ideals in n=3 variables, and the weak Lefschetz property for ideals in n=4 variables.