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arXiv · 1008.2377

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

Brian Harbourne·Hal Schenck·Alexandra Seceleanu
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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.

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Brian Harbourne, Hal Schenck, Alexandra Seceleanu. 2011-07-04. Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property. https://doi.org/10.1112/jlms%2Fjdr033

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