Combinatorially equivalent hyperplane arrangements
Introduction
Let V be a vector space of dimension l over a field K. Fix a system of coordinates of . We denote by the symmetric algebra of . A hyperplane arrangement 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 is a fundamental combinatorial invariant of an arrangement . 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 and 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 -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 in is called a hyperplane arrangement. For each hyperplane we fix a polynomial such that , and let . An arrangement is called central if each contains the origin of . In this case, each is a linear homogeneous polynomial, and hence is homogeneous of degree n.
Define the lattice of intersections of by where if , we identify with . We endow
Combinatorial equivalence
The results in this section are a generalization of certain ones from [20]. Fix a pair with and . Let be the set of affine arrangements of n distinct linearly ordered hyperplanes in . In other words, each element of is a collection , where are distinct affine hyperplanes in .
Definition 3.1 Given , define where and .
The space allows us to
Modular case
From now on we will assume that is a central and essential arrangement in . After clearing denominators, we can suppose that for all , and hence that . Moreover, we can also assume that there exists no prime number p that divides any .
Let p be a prime number, and consider the canonical homomorphism Since is central and we assume that there exists no prime number p that divides any , this implies that
On Terao's conjecture
We first recall the basic notions and properties of free hyperplane arrangements.
We denote by the S-module of polynomial vector fields on (or S-derivations). Let . Then δ is said to be homogeneous of polynomial degree d if are homogeneous polynomials of degree d in S. In this case, we write .
Let be a central arrangement in . Define the module of vector fields logarithmic tangent to (or logarithmic vector fields) by
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 . If for some , then p is not σ-lucky for the ideal . Proof By construction and are distinct homogenous polynomials of degree 1, that are not one multiple of the other. This implies that there exist two homogenous polynomials of degree 1 that form a minimal strong σ-Gröbner basis for . Notice that in this
On the period of arrangements
Let be a central and essential arrangement in , with for all . Moreover, assume that there exists no prime number p that divides any . We can associate to a integer matrix consisting of column vectors , for , such that Similarly, for each non-empty , we consider the integer matrix
For each prime number p, we can consider and 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).
References (21)
Characteristic polynomials of subspace arrangements and finite fields
Adv. Math.
(1996)- et al.
Subspace arrangements over finite fields: cohomological and enumerative aspects
Adv. Math.
(1997) On lucky ideals for Gröbner basis computations
J. Symb. Comput.
(1992)- et al.
Lefschetz properties and hyperplane arrangements
J. Algebra
(2020) - et al.
Signed graphs and freeness of the Weyl subarrangements of type
Discrete Math.
(2019) Moduli space of combinatorially equivalent arrangements of hyperplanes and logarithmic Gauss–Manin connections
Topol. Appl.
(2002)- et al.
An Introduction to Gröbner Bases
(1994) Computing the Tutte polynomial of a hyperplane arragement
Pac. J. Math.
(2007)Extended linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley
J. Algebraic Comb.
(1999)Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes
Bull. Lond. Math. Soc.
(2004)