  Let A be an integer (d x n) matrix, and assume that the convex hull conv(A)
of its columns is a simplex of dimension d-1. Write \NA for the semigroup
generated by the columns of A. It was proved by M. Saito [math.AG/0012257] that
the semigroup ring \CC[\NA] over the complex numbers \CC is Cohen-Macaulay if
and only if the rank of the GKZ hypergeometric system H_A(beta) equals the
normalized volume of conv(A) for all complex parameters beta in \CC^d. Our
refinement here shows, in this simplicial case, that H_A(beta) has rank
strictly larger than the volume of conv(A) if and only if beta lies in the
Zariski closure in \CC^d of all \ZZ^d-graded degrees where the local cohomology
H^i_m(\CC[\NA]) at the maximal ideal m is nonzero for some i < d.
