2008
DOI: 10.4153/cjm-2009-019-1
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The Geometry of the Weak Lefschetz Property and Level Sets of Points

Abstract: In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets … Show more

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Cited by 7 publications
(6 citation statements)
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“…A broad and interesting problem in this area is to determine structural results on the Hilbert functions of artinian algebras enjoying any of the three properties above: See, for instance, [8,12,13,16,19], or for the same problem with a more specific focus on special classes of algebras, such as Gorenstein or level, [2,3,6,7,10,11,14,17,18,19].…”
Section: Let a =mentioning
confidence: 99%
“…A broad and interesting problem in this area is to determine structural results on the Hilbert functions of artinian algebras enjoying any of the three properties above: See, for instance, [8,12,13,16,19], or for the same problem with a more specific focus on special classes of algebras, such as Gorenstein or level, [2,3,6,7,10,11,14,17,18,19].…”
Section: Let a =mentioning
confidence: 99%
“…If Z is a set of double points with general support and L is a general point with m = 2, the celebrated theorem of Alexander and Hirschowitz [2] gives a complete characterization of when equation 1 fails to give the correct count. Where m = 1 and L ⊆ P 3 is a line, the paper [21] relates equation 1 to Lefschetz properties of general Artinian reductions of R/I(Z).…”
Section: Introductionmentioning
confidence: 99%
“…We show that the general hyperplane section of this set has the weak Lefschetz property in almost every characteristic, whereas a special hyperplane section never has the weak Lefschetz property (see Corollary 3.8). Notice that examples of level sets of points in P 3 of type three such that every Artinian reduction fails the weak Lefschetz property have been constructed in [13,Section 3].…”
Section: Introductionmentioning
confidence: 99%