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

Migliore, Juan

doi:10.4153/cjm-2009-019-1

Cited by 7 publications

(6 citation statements)

References 24 publications

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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%

Forcing the strong Lefschetz and the maximal rank properties

Zanello

Zylinski

2009

Journal of Pure and Applied Algebra

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a b s t r a c tThree basic properties which standard graded artinian k-algebras may or may not enjoy are the Weak and Strong Lefschetz Properties and the Maximal Rank Property (respectively WLP, SLP, and MRP).In this paper we will assume that the base field k has characteristic zero. It is known that SLP implies MRP, which in turn implies WLP, but that both implications are strict. However, it surprisingly turned out (see [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for artinian K -algebras, J. Algebra 262 (2003) 99-126]) that the set of Hilbert functions admitting any algebras with WLP coincides with the corresponding set for SLP (and therefore with that for MRP). In [J. Migliore, F. Zanello, The Hilbert functions which force the weak Lefschetz property, J. Pure Appl. Algebra 210 (2) (2007) 465-471], J. Migliore and the first author, using Green's theorem and a result of Wiebe, characterized the Hilbert functions forcing all algebras to enjoy WLP. The purpose of this note is to prove the corresponding characterizations for both SLP and MRP. Unsurprisingly (or surprisingly??), the two characterizations coincide, but they define a class of Hilbert functions strictly smaller than that determined for WLP in the above cited reference.Our methods include the Herzog-Popescu theorem on quotients of k-algebras modulo a general form, a result of Wiebe, and gins and stable ideals. At the end, we will also discuss the importance of assuming that the characteristic be zero, and we will exhibit a class of codimension 2 monomial complete intersections for which SLP (but not MRP) fails in positive characteristic.

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Section: Let a =mentioning

confidence: 99%

Forcing the strong Lefschetz and the maximal rank properties

Zanello

Zylinski

2009

Journal of Pure and Applied Algebra

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“…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%

Projective Duality, Unexpected Hypersurfaces and Logarithmic Derivations of Hyperplane Arrangements

Trok

2020

Preprint

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Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal of Z. In this paper, we introduce the concept of a very unexpected hypersurface. This is a stronger condition which takes into account an explanation for some hypersurfaces previously considered unexpected.We then develop a duality theory to relate the study of very unexpected hypersurfaces to the derivations of dual hyperplane arrangements. This allows us to generalize results in the plane of Cook, Harbourne, Migliore, Nagel [7] Faenzi, and Vallés [13] to higher dimensions. In particular, we give a criterion to determine if a set Z admits very unexpected hypersurfaces in the case Z is invariant under the action of an irreducible reflection group on the ambient projective space.Our approach has new applications even in P 2 where we are able to place strong conditions on sets of points Z which admit certain types of unexpected curves. We close relating Terao's Freeness Conjecture for line arrangements to a conjecture due to G. Dirac [9] on configurations of points in R 2 .

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“…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%

Hyperplane sections and the subtlety of the Lefschetz properties

Cook

Nagel²

2012

Journal of Pure and Applied Algebra

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The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the property that a general hyperplane section has the weak Lefschetz property in almost all characteristics, whereas a special hyperplane section does not have this property in any characteristic.Comment: 9 page

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

Cited by 7 publications

References 24 publications

Forcing the strong Lefschetz and the maximal rank properties

Forcing the strong Lefschetz and the maximal rank properties

Projective Duality, Unexpected Hypersurfaces and Logarithmic Derivations of Hyperplane Arrangements

Hyperplane sections and the subtlety of the Lefschetz properties

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