### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*European Journal of Mathematics*,

*2*(4), 917-928. https://doi.org/10.1007/s40879-016-0109-1

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

Research output: Contribution to journal › Article

*European Journal of Mathematics*, vol. 2, no. 4, pp. 917-928. https://doi.org/10.1007/s40879-016-0109-1

}

TY - JOUR

T1 - Families of disjoint divisors on varieties

AU - Bogomolov, Fedor

AU - Pirutka, Alena

AU - Silberstein, Aaron Michael

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

UR - http://www.scopus.com/inward/record.url?scp=84995607535&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84995607535&partnerID=8YFLogxK

U2 - 10.1007/s40879-016-0109-1

DO - 10.1007/s40879-016-0109-1

M3 - Article

AN - SCOPUS:84995607535

VL - 2

SP - 917

EP - 928

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 4

ER -