Critical Points and Resonance of Hyperplane Arrangements

Cohen, D. ; Denham, G. ; Falk, M. ; Varchenko, A.

Canadian journal of mathematics, Tome 63 (2011) no. 5, pp. 1038-1057

Résumé

If ${{\Phi }_{\lambda }}$ is a master function corresponding to a hyperplane arrangement $\mathcal{A}$ and a collection of weights $\lambda $ , we investigate the relationship between the critical set of ${{\Phi }_{\lambda }}$ , the variety defined by the vanishing of the one-form ${{\omega }_{\lambda }}=\text{d}\log {{\Phi }_{\lambda }}$ , and the resonance of $\lambda $ . For arrangements satisfying certain conditions, we show that if $\lambda $ is resonant in dimension $p$ , then the critical set of ${{\Phi }_{\lambda }}$ has codimension at most $p$ . These include all free arrangements and all rank 3 arrangements.

Détail

DOI : 10.4153/CJM-2011-028-8

Mots-clés : 32S22, 55N25, 52C35, hyperplane arrangement, master function, resonant weights, critical set