On the weak Lefschetz property for powers of linear forms

Abstract

Ideals generated by prescribed powers of linear forms have attracted a great deal of attention recently. In this paper we study properties that hold when the linear forms are general, in a sense that we make precise. Analogously, one could study so-called “general forms” of the same prescribed degrees. One goal of this paper is to highlight how the differences between these two settings are related to the weak Lefschetz property (WLP) and the strong Lefschetz property (SLP). Our main focus is the case of powers of $r+1$ general linear forms in $r$ variables. For four variables, our results allow the exponents to all be different, and we determine when the WLP holds and when it does not in a broad range of cases. For five variables, we solve this problem in the case where all the exponents are equal (uniform powers), and in the case where one is allowed to be greater than the others. For evenly many variables ($\ge 6$) we solve the case of uniform powers, and in particular we prove half of a recent conjecture by Harbourne, Schenck and Seceleanu by showing that for evenly many variables, an ideal generated by $d$-th powers of $r+1$ general linear forms fails the WLP if and only if $d>1$. For uniform powers of an odd number of variables, we also give a result for seven variables, missing only the case $d=3$. Our approach in this paper is via the connection (thanks to Macaulay duality) to fat point ideals, together with a reduction to a smaller projective space, and the use of Cremona transformations.