On the strong Lefschetz question for uniform powers of general linear forms in $k[x,y,z]$

TL;DR

This paper investigates the strong Lefschetz property for uniform powers of general linear forms in three variables, proving maximal rank for square powers and analyzing behavior for higher powers in almost complete intersections.

Contribution

It provides the first systematic study of the strong Lefschetz property for uniform powers of linear forms, establishing maximal rank for square powers and detailed behavior for higher powers.

Findings

  imes L^2 always has maximal rank.

Behavior for higher powers depends on the exponent and the power, with at most one failure degree.

Higher powers of L fail maximal rank in at least two degrees, based on experimental evidence.

Abstract

Schenck and Seceleanu proved that if $R = k[x,y,z]$, where $k$ is an infinite field, and $I$ is an ideal generated by any collection of powers of linear forms, then multiplication by a general linear form $L$ induces a homomorphism of maximal rank from any component of $R/I$ to the next. That is, $R/I$ has the {\em weak Lefschetz property}. Considering the more general {\em strong Lefschetz question} of when $\times L^j$ has maximal rank for $j \geq 2$, we give the first systematic study of this problem. We assume that the linear forms are general and that the powers are all the same, i.e. that $I$ is generated by {\em uniform} powers of general linear forms. We prove that for any number of such generators, $\times L^2$ always has maximal rank. We then specialize to almost complete intersections, i.e. to four generators, and we show that for $j = 3,4,5$ the behavior depends on the uniform exponent and on $j$, in a way that we make precise. In particular, there is always at most one degree where $\times L^j$ fails maximal rank. Finally, we note that experimentally all higher powers of $L$ fail maximal rank in at least two degrees.