Matroid Representations and Free Arrangements

We show that Terao's Conjecture ("Freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid") holds over fields with at most four elements. However, an example demonstrates that the field characteristic has to be fixed for this. 1. Free arrangements The present study continues an investigation of the connection between algebraic and combinatorial structures of hyperplane arrangements [15, 17, 23, 24]. Specifically, the question ("Terao's Conjecture") is studied whether freeness of the module of logarithmic vector fields at an arrangement is determined by the underlying matroid. Here we apply representation theory for matroids to give positive answers in several important cases (binary matroids, arrangements over GF(2), GF(3), and GF(4)) where the arrangements are essentially projectively unique. However, it is shown that freeness of the arrangements corresponding to certain matroids does depend on the field characteristic: freeness cannot be recognized from the matroid alone. For this the technique of supersolvable resolutions is introduced, which allows freeness proofs (and disproofs) that depend on the embedding of the arrangement and thus on the specific representation of the underlying matroid. This leaves the feeling that Terao's Conjecture may well be wrong, although for several reasons hard to disprove: on the one hand freeness is a "rigidity property", and if a free arrangement is so rigid that it is projectively unique, then it cannot give rise to a counterexample; on the other hand the insight seems to emerge that the only arrangements that, combinatorially and philosophically speaking, have "a right to be free" (in the sense of Terao) are the supersolvable ones. Our hope is that the present note provides new insight into the combinatorial structure governing the algebraic properties of hyperplane arrangements. Received by the editors September 16, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 05B35, 32C40; Secondary 13H15, 51D25, 51M20. © 1990 American Mathematical Society 0002-9947/90 $1.00+ $.25 per page 525 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use