  Following the work of Totaro and Pereira, we study sufficient conditions
under which collections of pairwise-disjoint divisors on a variety over an
algebraically closed field are contained in the fibers of a morphism to a
curve. We prove that $\rho_w(X) + 1$ pairwise-disjoint, connected divisors
suffices for proper, normal varieties $X$, where $\rho_w(X)$ is a modification
of the N\'eron-Severi rank of $X$ (they agree when $X$ is projective and
smooth). We then prove a strong counterexample in the affine case: if $X$ is
quasi-affine and of dimension $\geq 2$ over a countable, algebraically-closed
field $k$, then there exists a (countable) collection of pairwise-disjoint
divisors which cover the $k$-points of X, so that for any non-constant morphism
from $X$ to a curve, at most finitely many are contained in the fibers thereof.
We show, however, that an uncountable collection of pairwise-disjoint,
connected divisors in any normal variety over an algebraically-closed field
must be contained in the fibers of a morphism to a curve.
