Balanced Configurations of Lattice Vectors and GKZ-rational Toric Fourfolds in P^6

Abstract

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,R) equivalence and deduce that the only gkz-rational toric four-folds in complex projective space P^6 are those varieties associated with an essential Cayley configuration. In this case, we study a suitable hyperplane arrangement and show that all rational A-hypergeometric functions may be described in terms of toric residues.Comment: Revised version to appear in the Journal of Algebraic Combinatorics. The new proof of Theorem 2.14 is inspired by the proof of N. Ressayre in math.RA/0206234 of a conjecture we raised in the previous version of this articl