  The holonomic rank of an A-hypergeometric system $H_A(\beta)$ is conjectured
to be independent of the parameter vector $\beta$ if and only if the toric
ideal $I_A$ is Cohen Macaulay. We prove this conjecture in the case that $I_A$
is generic by explicitly constructing more than $\vol(A)$ many linearly
independent hypergeometric functions for parameters $\beta$ coming from
embedded primes of certain initial ideals of $I_A$.
