Lefschetz property and powers of linear forms in 𝕂[ x , y , z ]

Tran

2020

Journal of Algebra

In [11], Conjecture 6.6, Migliore, the first author, and Nagel conjectured that, for all n ≥ 4, the artinian ideal. , x 2n ] generated by the d-th powers of 2n + 2 general linear forms fails to have the weak Lefschetz property if and only if d > 1. This paper is entirely devoted to prove partially this conjecture. More precisely, we prove that R/I fails to have the weak Lefschetz property, provided 4 ≤ n ≤ 8, d ≥ 4 or d = 2r, 1 ≤ r ≤ 8, 4 ≤ n ≤ 2r(r + 2) − 1.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms

Tran

2020

Journal of Algebra

“…There are plenty of results concerning Lefschetz properties of ideals generated by powers of linear forms (see e.g., [2,9,11,13]), and it is natural to consider generalizations of those results to ideals generated by products of linear forms. From this point of view, it would be interesting to study Question 1.1 for artinian complete intersections generated by products of linear forms.…”

Section: Introductionmentioning

confidence: 99%

Lefschetz properties for complete intersection ideals generated by products of linear forms

Juhnke-Kubitzke

Murai

et al. 2018

Proc. Amer. Math. Soc.

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

Migliore

2017

Proc. Amer. Math. Soc.

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 question of when ×L j 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 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 ×L j fails maximal rank. Finally, we note that experimentally all higher powers of L fail maximal rank in at least two degrees.which proves what we want.Remark 6.2. It is natural to ask what happens for ideals generated by uniform powers of more than four general linear forms. Here is what happens for 6 general linear forms, at least experimentally using [2].Here, ×L j fails to have maximal rank in all degrees for the following values of k (taking 3 ≤ j ≤ 10, 3 ≤ k ≤ 30).