Harbourne, Schenck and Seceleanu's Conjecture

Abstract

Abstract. In [2], Conjecture 5.5.2, Harbourne, Schenck and Seceleanu conjectured that, for r = 6 and all r ≥ 8, the artinian ideal. . , xr] generated by the square of r + 1 general linear forms ℓi fails the Weak Lefschetz property. This paper is entirely devoted to prove this Conjecture. It is worthwhile to point out that half of the Conjecture -namely, the case when the number of variables r is even -was already proved in [5], Theorem 6.1.

Interpolation and the weak Lefschetz property

“…• If n = 6 then R/I has the weak Lefschetz property if and only if d ∈ {1, 2} (see [7] for d = 3 and [29] for d = 3) . • If n ≥ 8 is even then R/I has the weak Lefschetz property if and only if d = 1 (known for d = 2 by [26]). The last item is a restatement of Conjecture 6.1.…”

“…A graded Artinian algebra A is said to have the weak Lefschetz property (WLP) if it has a linear form ℓ such that multiplication by ℓ on A has maximal rank from each degree to the next. Deciding if an algebra has the WLP is often a delicate problem, and there is a rich literature on this topic (see, e.g., [5,7,20,26,34,37,38]). Of particular interest is the case, where A = R/I and the ideal I is generated by powers of general linear forms (see, e.g., [19,29,27,30]).…”

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

“…One might suspect that Problem 4.1 generalizes to almost complete intersection ideals generated by quadrics, which are products of linear forms. However, in [11,Theorem 2.12], the second author proved that this is not the case. Indeed, for n = 6 and all n ≥ 8, the artinian ideal I = (ℓ 2 1 , .…”

“…As an extension of the previous problem, we propose the following problem: Concerning this last problem it is worthwhile to point out that there is a huge list of papers dealing with ideals generated by powers of linear forms. For more information on this subject the reader can see, for instance, [4], [9], [11] and [13]. The problem is really subtle since a minuscule change can alter the behavior of the WLP.…”

“…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.…”