2008
DOI: 10.4153/cjm-2008-019-2
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The Geometry of the Weak Lefschetz Property and Level Sets of Points

Abstract: 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 l… Show more

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Cited by 9 publications
(6 citation statements)
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“…As an immediate corollary, since monomial artinian ideals lift to ideals of points, we have another proof of the existence of reduced level sets of points in P r (for every r ≥ 3) whose artinian reductions are non-unimodal (this result was first shown in [Mi,Theorem 4.3]). …”
Section: Introductionmentioning
confidence: 82%
“…As an immediate corollary, since monomial artinian ideals lift to ideals of points, we have another proof of the existence of reduced level sets of points in P r (for every r ≥ 3) whose artinian reductions are non-unimodal (this result was first shown in [Mi,Theorem 4.3]). …”
Section: Introductionmentioning
confidence: 82%
“…In [53], an example is constructed of a level set of points in P 3 with the property that an Artinian reduction (hence any Artinian reduction) has a non-unimodal Hilbert function, but it is remarked that no example is known of more than two maxima. (By this we mean that there is at most one "valley" in the sense of [7].)…”
Section: Moreover If Equality Holds In Some Degree J Then It Holds mentioning
confidence: 99%
“…In fact, he proved a stronger result than unimodality using the structure theorem of Buchsbaum and Eisenbud for the Gorenstein algebra of codimension 3 in [8]. Since then, the graded Artinian Gorenstein algebras of codimension 3 have been much studied (see [9,15,16,20,21,27,28,31,33]). In [3], Bernstein and Iarrobino showed how to construct non-unimodal graded Artinian Gorenstein algebras of codimension higher than or equal to 5.…”
Section: Introductionmentioning
confidence: 97%