Product of Hessians and Discriminant of Critical Points of Level Function for Hypergeometric Integrals

Abstract

We give in two examples (hyperplane arrangement and circle arrangement) a relation between the product of the Hessians of a level function associated with hypergeometric integrals and discriminant attached to them. We also express the product of values of prime factors of integrand at critical points by the use of basic invariants attached to each arrangement. Our main goal is to give a criterion in terms of the product of Hessians for that all critical points are different from each other.