  In this paper we answer a question posed by V.V. Batyrev. The question asked
if there exists a complete regular fan with more than quadratically many
primitive collections. We construct a smooth projective toric variety
associated to a complete regular fan $\Delta$ in R^d with $n$ generators where
the number of primitive collections of $\Delta$ is at least exponential in
$n-d$. We also exhibit the connection between the number of primitive
collections of $\Delta$ and the facet complexity of the Gr\"obner fan of the
associated integer program.
