Families of Disjoint Divisors on Varieties

Fedor A. Bogomolov; Alena Pirutka; Aaron Michael Silberstein

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arXiv · 1504.05534

Families of Disjoint Divisors on Varieties

Fedor A. Bogomolov·Alena Pirutka·Aaron Michael Silberstein

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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 $ρ_w(X) + 1$ pairwise-disjoint, connected divisors suffices 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 $\geq 2$ 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.

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BibTeX RIS

Fedor A. Bogomolov, Alena Pirutka, Aaron Michael Silberstein. 2016-01-20. Families of Disjoint Divisors on Varieties. https://arxiv.org/abs/1504.05534

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