Families of disjoint divisors on varieties
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- 作者：Fedor A. Bogomolov ; Alena Pirutka…
- 关键词：Disjoint divisors ; Regular functions ; Hodge index theorem ; Chow group ; Geometric reconstruction
- 刊名：European Journal of Mathematics
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：2
- 期：4
- 页码：917-928
- 全文大小：423 KB
- 刊物类别：Algebraic Geometry;
- 刊物主题：Algebraic Geometry;
- 出版者：Springer International Publishing
- ISSN：2199-6768
- 卷排序：2
文摘
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 \(\rho _\mathrm{w}(X) + 1\) pairwise-disjoint, connected divisors suffice for proper, normal varieties X, where \(\rho _\mathrm{w}(X)\) 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 \(\geqslant \)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.