TY - JOUR

T1 - Families of disjoint divisors on varieties

AU - Bogomolov, Fedor A.

AU - Pirutka, Alena

AU - Silberstein, Aaron Michael

N1 - Funding Information:
The first author acknowledges that the article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program. The first author was also supported by a Simons Travel Grant. The second author thanks the University of Chicago and the Embassy of France in United States for their support for a short-term visit to the University of Chicago in April, 2015. The third author gratefully acknowledges the support of NSF Grant DMS-1400683. This paper was conceived at Workshop # 1444 at Oberwolfach, and we gratefully acknowledge the hospitality of the Matematisches Forschungsinstitut Oberwolfach.
Publisher Copyright:
© 2016, Springer International Publishing AG.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - Chow group

KW - Disjoint divisors

KW - Geometric reconstruction

KW - Hodge index theorem

KW - Regular functions

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U2 - 10.1007/s40879-016-0109-1

DO - 10.1007/s40879-016-0109-1

M3 - Article

AN - SCOPUS:84995607535

SN - 2199-675X

VL - 2

SP - 917

EP - 928

JO - European Journal of Mathematics

JF - European Journal of Mathematics

IS - 4

ER -