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

Abstract

Abstract. We show that an Artinian quotient of an ideal I ⊆ K[x, y, z] generated by powers of linear forms has the Weak Lefschetz Property. If the syzygy bundle of I is semistable, the property follows from results of Brenner-Kaid. Our proof works without this hypothesis, which typically does not hold.

Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

“…Results of Harima, Migliore, Nagel and Watanabe [13] show that WLP always holds for r 2. For r = 3, WLP holds for ideals of general forms [2], complete intersections [13], ideals with semistable syzygy bundle and certain splitting type, and almost complete intersections with unstable syzygy bundle [3], certain monomial ideals [18] and ideals generated by powers of linear forms [21]. The following example of Migliore-Miró-Roig-Nagel [18] shows that the result of Schenck and Seceleanu [21] can fail for r 4, and motivates this paper.…”

“…This is surprising: for I ⊆ K[x 1 , x 2 , x 3 , x 4 ] generated by general forms, Migliore and Miró-Roig show in [16] that the quotient ring always has WLP. It also contrasts to most known cases of powers of linear forms: WLP always holds in the three-variable case [21] and for complete intersections (that is, r = n). The result on complete intersections is due to Stanley [22], who showed that if I = l t1 1 , .…”

“…Scripts to analyze the WLP are available at: http://www.math.uiuc.edu/∼asecele2. We thank Pietro Majer for bringing equation (21) to our attention, Juan Migliore for pointing out a gap in an earlier version and the referee for many useful comments.…”

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

“…1 has maximal rank for all i, s. When this property holds, the algebra is said to have the strong Lefschetz property (briefly SLP). In [14], Schenck and Seceleanu gave the nice result that any artinian ideal I ⊂ R = k[x, y, z] generated by powers of linear forms has the WLP. Moreover, when these linear forms are general, the SLP of R/I has also been studied, in particular, the multiplication by the square ℓ 2 of a general linear form ℓ induces a homomorphism of maximal rank in any graded component of R/I, see [1,10].…”

“…If d = 4, then D = 0, by Proposition 3.2. With the notations as in the case 2, one has L 11 (11; 914 ) = dim[R 6,3 ] 11 if d = 5 dim k L 11 (10; 8 14 ) = dim[R 6,3 ] 10 if d = 7 dim k L 11 (9; 7 14 ) = dim[R 6,3 ] 9 if d = 9.…”