Families of disjoint divisors on varieties

Fedor Bogomolov, Alena Pirutka, Aaron Michael Silberstein

Research output: Contribution to journalArticle

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 ρw(X) + 1 pairwise-disjoint, connected divisors suffice for proper, normal varieties X, where ρw(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.

Original languageEnglish (US)
Pages (from-to)917-928
Number of pages12
JournalEuropean Journal of Mathematics
Volume2
Issue number4
DOIs
StatePublished - Dec 1 2016

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Divisor
Pairwise
Disjoint
Morphism
Algebraically closed
Fiber
Curve
Countable
Uncountable
Counterexample
Cover
Family
Sufficient Conditions

Keywords

  • Chow group
  • Disjoint divisors
  • Geometric reconstruction
  • Hodge index theorem
  • Regular functions

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Families of disjoint divisors on varieties. / Bogomolov, Fedor; Pirutka, Alena; Silberstein, Aaron Michael.

In: European Journal of Mathematics, Vol. 2, No. 4, 01.12.2016, p. 917-928.

Research output: Contribution to journalArticle

Bogomolov, Fedor ; Pirutka, Alena ; Silberstein, Aaron Michael. / Families of disjoint divisors on varieties. In: European Journal of Mathematics. 2016 ; Vol. 2, No. 4. pp. 917-928.
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