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arXiv · 2610.34517

Generators and Lefschetz properties of general hyperplane arrangement apolar algebras

Jarosław Buczyński·Zach Teitler·Alexander Woo
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Abstract

Hyperplane arrangements are rich combinatorial, topological, and geometric objects with numerous applications. Apolarity is a method of turning a homogeneous polynomial into a finite graded Gorenstein algebra. Invariants of the apolar algebra of the defining equation of a hyperplane arrangement are invariants of the arrangement itself. We investigate such invariants, including the Hilbert function, minimal generators of the apolar ideal, and Lefschetz properties. The main focus of the article is on combinatorially general arrangements of degree d in dimension n with d between n and 2n-2. In this range, we determine the Hilbert function and conjecture that the same formula is valid for larger values of d too. Minimal generators of the apolar ideal in the initial degree are in natural bijection to the lines of the arrangement. The initial degree characterizes arrangements that are combinatorially general. Finally, we prove that the apolar algebra has the Weak Lefschetz Property and precisely characterize the Weak Lefschetz elements.

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Jarosław Buczyński, Zach Teitler, Alexander Woo. 2026-10-25. Generators and Lefschetz properties of general hyperplane arrangement apolar algebras. https://arxiv.org/abs/2610.34517

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