New characterizations of freeness for hyperplane arrangements

Abstract

In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements. DI

Lefschetz properties and hyperplane arrangements

“…Notice that Terao described this result for characteristic 0, but the statement holds true for any characteristic as shown in [10]. In [4], the authors connected the study of generic initial ideals to the one of arrangements, obtaining a new characterization of freeness via the generic initial ideal of the Jacobian ideal. with 1 ≤ λ 1 < λ 2 < · · · < λ n−1 and λ i+1 − λ i = 1 or 2.…”

Hyperplane arrangements in CoCoA

“…In fact, freeness implies several interesting geometric and combinatorial properties of the arrangement itself. See for example [18], [20], [4], [7] and [12].…”

Untitled

“…Difference sets were first studied in relation to finite geometries [Singer 1938] and have connections to symmetric designs, coding theory, and many other fields of mathematics [Moore and Pollatsek 2013;Davis and Jedwab 1996;Colbourn and Dinitz 1996;Beth et al 1999].…”

“…Large libraries of difference sets are useful for developing conjectures and building examples. Gordon provides an extensive library of difference sets in abelian groups [Gordon], but has no results for nonabelian groups, which do show distinct behavior [Smith 1995]. A wide variety of techniques can be used to construct difference sets for these libraries (see, for example, [Dillon 1985] and [Davis and Jedwab 1997]), but fully enumerating all difference sets in a given group requires some amount of exhaustive search, which can quickly become computationally infeasible.…”

“…First, since every difference set is equivalent to some difference set containing the identity, the algorithm does not enumerate some preimages that are guaranteed to be equivalent to others. Second, the final elimination of all but one representative of equivalence classes of difference sets uses the SmallestImageSet function [Linton 2004] from the GAP package [GRAPE]. Although roughly 20% slower than the function given above for most cases, SmallestImageSet gives a unique minimal result and handles groups with large automorphism groups much more efficiently.…”