Families of Disjoint Divisors on Varieties

Abstract

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 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 $\ge 2$ 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.

arXiv commentAdded new author (A.P.), reworked and strengthened arguments

References in corpus (1)

• Étale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields