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

Migliore, Juan C.

Canadian journal of mathematics, Tome 60 (2008) no. 2, pp. 391-411

Résumé

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.

Détail

DOI : 10.4153/CJM-2008-019-2

Mots-clés : 13D40, 13D02, 14C20, 13C40, 13C13, 14M05, Weak Lefschetz Property, Strong Lefschetz Property, basic double G-linkage, level, arithmetically Gorenstein, arithmetically Cohen–Macaulay, socle type, socle degree, Artinian reduction

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