Free arrangements and relation spaces
For \(\{x_ 1, \dots, x_ n\}\) a basis of the dual vector \(n\)-space \((V^ n)^*\) and \(S\) a symmetric algebra of \((V^ n)^*\), isomorphic to the polynomial algebra \(\mathbb{K} [x_ 1, \dots, x_ n]\), each hyperplane \(H \subset V^ n\) has a defining form \(\alpha_ H = a_ 1x_ 1 + \cdots + a_ nx_ n\) with \(\text{ker} (\alpha_ H) = H\) (unique up to a constant). Thus, an arrangement \(A\) of hyperplanes in \(V^ n\) can be described by the product of such forms. So let \(Q = Q(A) = \prod_{H \in A} \alpha_ H\), and let \(D(A)\) be the \(S\)-module consisting of all deviations \(\Theta : S \to S\) such that \(\Theta (Q)\) is a multiple of \(Q\). If \(D(A)\) is a free \(S\)-module, then the arrangement \(A\) is said to be free. On the other hand, \(A\) is formal if all linear dependencies among the defining forms of the hyperplanes of \(A\) are generated by dependencies corresponding to the 2-codimensional subspaces. \textit{S. Yuzvinsky} [Trans. Am. Math. Soc. 335, No. 1, 231-244 (1993; Zbl 0768.05019)] proved that free arrangements are formal. In the present paper it is shown that if \(A\) is free, then \(A\) is even \(k\)-formal (with \(2 \leq k \leq r - 1\), where \(r\) is the codimension of the common intersection of all the hyperplanes from \(A)\). For this assertion, the more general \(k\)-formal arrangements are introduced, where, as starting point, \(A\) is 2-formal if it is formal. (The authors construct also a 3- formal arrangement which is not 2-formal, to distinguish the notion ``\(k\)-formal'' from the notion ``formal'').
scientific article; zbMATH DE number 617324
scientific article