## 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 suffice 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 ⩾ 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 language | English (US) |
---|---|

Pages (from-to) | 917-928 |

Number of pages | 12 |

Journal | European Journal of Mathematics |

Volume | 2 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2016 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)