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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 weak Lefschetz property. Considering the more general strong Lefschetz problem of when ×L has maximal rank for j ≥ 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 uniform powers of general linear forms. We prove that for any number of such generators, ×L 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 ×L fails maximal rank. Finally, we note that experimentally all higher powers of L fail maximal rank in at least two degrees.

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