On the Complexity of Smooth Projective Toric Varieties

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.