Families of Disjoint Divisors on Varieties
Fedor A. Bogomolov, Alena Pirutka, and Aaron Michael Silberstein

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 pairwise-disjoint, connected divisors suffices for proper, normal varieties , where is a modification of the N\'eron-Severi rank of (they agree when is projective and smooth). We then prove a strong counterexample in the affine case: if is quasi-affine and of dimension over a countable, algebraically-closed field , then there exists a (countable) collection of pairwise-disjoint divisors which cover the -points of X, so that for any non-constant morphism from to a curve, at most finitely many are contained in the fibers thereof. We show, however, that an uncountable collection of…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Commutative Algebra and Its Applications
