[Paper Review] Deletion theorem and combinatorics of hyperplane arrangements

Takuro Abe|arXiv (Cornell University)|Sep 18, 2017

Advanced Combinatorial Mathematics13 references3 citations

TL;DR

This paper establishes that the deletion theorem for free hyperplane arrangements is combinatorial: whether removing a hyperplane preserves freeness depends only on the intersection lattice. The key result is a precise condition using characteristic polynomials—specifically, that the restriction’s characteristic polynomial divides the original’s for all relevant flats—providing a purely combinatorial criterion for freeness after deletion.

ABSTRACT

We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.

Motivation & Objective

• To determine whether the freeness of a hyperplane arrangement after deleting a hyperplane depends only on its combinatorial structure (the intersection lattice).
• To resolve a long-standing open problem in arrangement theory: whether the deletion theorem is combinatorial, particularly in relation to Terao’s conjecture on the combinatorial nature of freeness.
• To provide a sufficient and necessary condition for freeness after deletion in terms of characteristic polynomials, independent of algebraic or geometric data.
• To establish that the existence of free filtrations and the modified Orlik’s conjecture are also combinatorial properties.
• To extend the applicability of the $b_2$-equality and logarithmic derivation module structure to derive combinatorial criteria for freeness.

Proposed method

• Leverages the $b_2$-equality condition on multiarrangements to analyze the structure of logarithmic derivation modules.
• Applies a structure theorem for minimal generators of $D({ ilde{\mathcal{A}}}, m)$ when the $b_2$-equality holds, linking algebraic properties to combinatorics.
• Uses induction on the dimension $\ell$, reducing the problem to lower-dimensional restrictions and applying the induction hypothesis.
• Employs the Ziegler restriction and local freeness arguments to analyze the behavior of characteristic polynomials under restriction.
• Applies Terao’s addition-deletion theorem and the divisibility of characteristic polynomials to derive necessary and sufficient conditions.
• Uses exact sequences of sheaves and stalk-wise analysis at points in the support to derive contradictions when assumptions fail.

Experimental results

Research questions

• RQ1 Is the deletion theorem for free arrangements combinatorial, i.e., does the freeness of $\mathcal{A} \setminus \{H\}$ depend only on the intersection lattice $L(\mathcal{A})$?
• RQ2 Can the freeness of the restriction $\mathcal{A}^H$ be determined combinatorially when $\chi(\mathcal{A}^H; t)$ divides $\chi(\mathcal{A}; t)$?
• RQ3 Does the existence of a free filtration in a free arrangement depend only on its intersection lattice?
• RQ4 Is the modified Orlik’s conjecture—whether $\mathcal{A}^H$ is free when $\chi(\mathcal{A}^H; t) \mid \chi(\mathcal{A}; t)$—a combinatorial property?
• RQ5 What is the precise algebraic-combinatorial condition under which $\mathcal{A} \setminus \{H\}$ remains free, given that $\mathcal{A}$ is free?

Key findings

• The deletion theorem for free arrangements is combinatorial: $\mathcal{A} \setminus \{H\}$ is free if and only if $\chi(\mathcal{A}^H_X; t)$ divides $\chi(\mathcal{A}_X; t)$ for all $X \in L_i(\mathcal{A}^H)$ with $2 \leq i \leq \ell - 1$.
• This condition is both necessary and sufficient, and it depends only on the intersection lattice $L(\mathcal{A})$, making the freeness of the deletion combinatorial.
• The existence of free filtrations in free arrangements is a combinatorial property, as it reduces to iterated application of the deletion condition.
• For $\ell \leq 4$, if $\chi(\mathcal{A}^H; t)$ divides $\chi(\mathcal{A}; t)$, then $\mathcal{A}$ is free if and only if $\mathcal{A}^H$ is free.
• The modified Orlik’s conjecture is combinatorial: under the assumption $\chi(\mathcal{A}^H; t) \mid \chi(\mathcal{A}; t)$, the freeness of $\mathcal{A}^H$ depends only on $L(\mathcal{A})$.
• The $b_2$-equality condition on $H \in \mathcal{A}_X$ and the root condition $|\mathcal{A}_X| - |\mathcal{A}_X^H|$ being a root of $\chi(\mathcal{A}_X; t)$ are equivalent to the deletion theorem holding, providing an alternative combinatorial criterion.