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arXiv · 2406.09571

On the Weak Lefschetz Property for certain ideals generated by powers of linear forms

Giuseppe Favacchio·Juan Migliore
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Abstract

Ideals $I\subseteq R=k[\mathbb P^n]$ generated by powers of linear forms arise, via Macaulay duality, from sets of fat points $X\subseteq \mathbb P^n$. Properties of $R/I$ are connected to the geometry of the corresponding fat points. When the linear forms are general, many authors have studied the question of whether or not $R/I$ has the Weak Lefschetz Property (WLP). We study this question instead for ideals coming from a family of sets of points called grids. We give a complete answer in the case of uniform powers of linear forms coming from square grids, and we give a conjecture and approach for the case of nonsquare grids. In the cases where WLP holds, we also describe the non-Lefschetz locus.

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Giuseppe Favacchio, Juan Migliore. 2024-06-13. On the Weak Lefschetz Property for certain ideals generated by powers of linear forms. https://arxiv.org/abs/2406.09571

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