Combinatorially equivalent hyperplane arrangements

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Abstract

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong σ-Gröbner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.

Introduction

Let V be a vector space of dimension l over a field K. Fix a system of coordinates (x1,,xl) of V. We denote by S=S(V)=K[x1,,xl] the symmetric algebra of V. A hyperplane arrangement A={H1,,Hn} is a finite collection of hyperplanes in V. For a thorough treatment of the theory of hyperplane arrangements and recent developments, see [13], [16], [10] and [17].

The lattice of intersections L(A) is a fundamental combinatorial invariant of an arrangement A. In fact one of the most studied topics in the theory of arrangements is to identify which topological and algebraic invariants of an arrangement are determined by its lattice of intersections.

To pursue this type of questions, Athanasiadis ([3], [4] and [5]), inspired by [9] and [8], initiated and systematically applied the “finite field method”, i.e. the study of the combinatorics of arrangements and their reduction modulo prime numbers. See also [7] for related work. After its introduction, this method has been used by several authors ([11], [12], [2] and [15]) to solve similar problems. The purpose of this paper is to study the combinatorics of arrangements over arbitrary fields and determine in which situation an arrangement and its reduction modulo a prime have isomorphic lattices.

The paper is organized as follows. In Section 2, we recall the basic notions on hyperplane arrangements. In Section 3, we describe how to characterize when two arrangements are combinatorially equivalent. In Section 4, we use the results of Section 3 to describe the primes p for which A and Ap are combinatorially equivalent. In Section 5, we show that the knowledge of Terao's conjecture in finite characteristic implies the conjecture over the rationals. In Section 6, we describe a method to compute good primes via minimal strong σ-Gröbner bases. In Section 7, we show that computing the good and (σ,l)-lucky primes for an arrangement is equivalent to compute all the primes that divide its lcm-period (as defined in [12]).

Section snippets

Preliminaries

Let K be a field. A finite set of affine hyperplanes A={H1,,Hn} in Kl is called a hyperplane arrangement. For each hyperplane Hi we fix a polynomial αiS=K[x1,,xl] such that Hi=αi1(0), and let Q(A)=i=1nαi. An arrangement A is called central if each Hi contains the origin of Kl. In this case, each αi is a linear homogeneous polynomial, and hence Q(A) is homogeneous of degree n.

Define the lattice of intersections of A byL(A)={HBH|BA}, where if B=, we identify HBH with Kl. We endow L(A)

Combinatorial equivalence

The results in this section are a generalization of certain ones from [20]. Fix a pair (l,n) with l1 and n0. Let An(Kl) be the set of affine arrangements of n distinct linearly ordered hyperplanes in Kl. In other words, each element A of An(Kl) is a collection A={H1,,Hn}, where H1,,Hn are distinct affine hyperplanes in Kl.

Definition 3.1

Given AAn(Kl), defineI(A¯)={(i1,,il+1)[n+1]<l+1|H¯i1H¯il+1}, where [n+1]={1,,n+1} and [n+1]<l+1={(i1,,il+1)[n+1]l+1|i1<<il+1}.

The space I(A¯) allows us to

Modular case

From now on we will assume that A={H1,,Hn} is a central and essential arrangement in Ql. After clearing denominators, we can suppose that αiZ[x1,,xl] for all i=1,,n, and hence that Q(A)=i=1nαiZ[x1,,xl]. Moreover, we can also assume that there exists no prime number p that divides any αi.

Let p be a prime number, and consider the canonical homomorphismπp:Z[x1,,xl]Fp[x1,,xl]. Since A is central and we assume that there exists no prime number p that divides any αi, this implies that πp(αi)

On Terao's conjecture

We first recall the basic notions and properties of free hyperplane arrangements.

We denote by DerKl={i=1lfixi|fiS} the S-module of polynomial vector fields on Kl (or S-derivations). Let δ=i=1lfixiDerKl. Then δ is said to be homogeneous of polynomial degree d if f1,,fl are homogeneous polynomials of degree d in S. In this case, we write pdeg(δ)=d.

Let A be a central arrangement in Kl. Define the module of vector fields logarithmic tangent to A (or logarithmic vector fields) byD(A)={δDerKl|

How to compute good primes via Gröbner bases

We will now describe a method to compute good primes for an arrangement using minimal strong σ-Gröbner bases.

Lemma 6.1

Let 1i<jn. If (αi)p=β(αj)p for some βFp{0}, then p is not σ-lucky for the ideal αi,αjZZ[x1,,xl].

Proof

By construction αi and αj are distinct homogenous polynomials of degree 1, that are not one multiple of the other. This implies that there exist g1,g2Z[x1,,xl] two homogenous polynomials of degree 1 that form a minimal strong σ-Gröbner basis for αi,αjZ. Notice that in this

On the period of arrangements

Let A={H1,,Hn} be a central and essential arrangement in Ql, with αiZ[x1,,xl] for all i=1,,n. Moreover, assume that there exists no prime number p that divides any αi. We can associate to A a l×n integer matrixC=(c1,,cn)Matl×n(Z) consisting of column vectors ci=(c1i,,cli)TZl, for i=1,,n, such thatαi=k=1lckixk. Similarly, for each non-empty J={i1,,ik}[n], we consider the l×k integer matrixCJ=(ci1,,cik)Matl×k(Z).

For each prime number p, we can consider (C)p and (CJ)p the reductions

Acknowledgments

The authors would like to thank M. Yoshinaga for many helpful discussions. During the preparation of this article the second author was supported by JSPS Grant-in-Aid for Early-Career Scientists (19K14493).

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