Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property

Abstract

Migliore-Miró-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation in K[x_1,x_2,x_3], where WLP always holds [H.Schenck, A.Seceleanu, Proc. A.M.S. 2010, arXiv:0911.0876]. We use the inverse system dictionary to connect I to an ideal of fat points and show that failure of WLP for powers of linear forms is connected to the geometry of the associated fat point scheme. Recent results of Sturmfels-Xu in [J. Eur. Math. Soc. 2010, arXiv:0803.0892] allow us to relate WLP to Gelfand-Tsetlin patterns. See the paper "On the weak Lefschetz property for powers of linear forms" by Migliore-Miró-Roig-Nagel [arXiv:1008.2149] for related results.

to appear in J. London Math. Soc

References in corpus (4)

• Syzygy Bundles on P^2 and the Weak Lefschetz Property
• Problems and progress: survey on fat points in P2
• On linear systems of P^3 through multiple points
• The Weak Lefschetz Property and powers of linear forms in K[x,y,z]

Cited by in corpus (12)

• A tour of the Weak and Strong Lefschetz Properties
• The weak Lefschetz property for Artinian Gorenstein algebras of codimension three
• On the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms
• Unexpected hypersurfaces and where to find them
• Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property
• The weak Lefschetz property of Gorenstein algebras of codimension three associated to the Apéry sets
• A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms
• On the strong Lefschetz question for uniform powers of general linear forms in $k[x,y,z]$
• On the Weak Lefschetz Property for Powers of Linear Forms
• Representation theory of symmetric groups and the strong Lefschetz property
• Lefschetz properties for complete intersection ideals generated by products of linear forms
• Finding Inverse Systems from Coordinates