Families of disjoint divisors on varieties

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 ρ_w(X) + 1 pairwise-disjoint, connected divisors suffice for proper, normal varieties X, where ρ_w(X) is a modification of the Néron–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 ⩾ 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.