Question: Let $n$ be a positive integer and let $X$ be a simplicial toric variety. Does every coset of $n\cdot Pic(X)\subseteq Pic(X)$ contain a base point free linear system?
If $X$ is not simplicial, the answer is definitely no, since there exist proper toric varieties with no non-trivial line bundles (for example, Example 4.2.13 of Cox, Little, and Schenck).
Obviously, if $X$ is quasi-projective, every coset of $n\cdot Pic(X)$ contains a very ample (so base point free) line bundle. So the question is only interesting for non-quasi-projective $X$.
I'll rephrase the question in terms of fans. Given a rational ray $\rho$, let $v_\rho$ denote the first lattice point along $\rho$.
Rephrasing: Let $n$ be a positive integer and let $\Sigma$ be a simplicial fan on a lattice $N$. Let $\{a_\rho\}_{\rho\in \Sigma(1)}$ be integers† associated to the rays of $\Sigma$. Do there exist integers† $\{b_\rho\}_{\rho\in \Sigma(1)}$ so that for every $\rho$, there is a point $\chi\in N^*$ so that $\chi(v_\rho) + a_\rho+nb_\rho=0$ and $\chi(v_{\rho'})+a_{\rho'}+nb_{\rho'}\ge 0$ for every $\rho'\in \Sigma(1)$?
† These sets of integers are assumed to be induced by integer-valued piecewise linear functions. That is, they correspond to Cartier divisors.
In other words, given a bunch of integral† hyperplanes perpendicular to the rays of $\Sigma$, is it possible to move each of them by a multiple of $n$ (†, sigh) so that each of them touches the polytope they define?
