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The Geometry of the Weak Lefschetz Property and Level Sets of Points

Published online by Cambridge University Press:  20 November 2018

Juan C. Migliore
Juan C. Migliore*
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. e-mail: juan.c.migliore.1@nd.edu
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Abstract

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In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property $(\text{WLP})$, 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 $\text{WLP}$, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have $\text{WLP}$. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has $\text{WLP}$ but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double $G$-linkage, the implementations use very different methods.

Keywords

13D4013D0214C2013C4013C1314M05Weak Lefschetz PropertyStrong Lefschetz Propertybasic double G-linkagelevelarithmetically Gorensteinarithmetically Cohen–Macaulaysocle typesocle degreeArtinian reduction
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 2 , 01 April 2008 , pp. 391 - 411
Copyright
Copyright © Canadian Mathematical Society 2008

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