Families of Disjoint Divisors on Varieties

TL;DR

This paper investigates conditions under which collections of disjoint divisors on varieties are contained in fibers of morphisms to curves, providing new bounds for proper varieties and counterexamples in affine cases.

Contribution

It establishes that a bound of  ho_w(X)+1 divisors suffices for proper, normal varieties and demonstrates a counterexample in the affine case, advancing understanding of divisor configurations.

Findings

 ho_w(X)+1 divisors suffice for proper, normal varieties

Counterexample exists for affine, quasi-affine varieties with disjoint divisors covering all points

Uncountable collections of disjoint divisors in normal varieties are contained in fibers of a morphism to a curve

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 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.