Hypersurfaces in projective schemes and a moving lemma

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain...

Full description

Saved in:
Bibliographic Details
Published in:Duke mathematical journal 2015, Vol.164 (7), p.1187-1270
Main Authors: Gabber, Ofer, Liu, Qing, Lorenzini, Dino
Format: Article
Language:English
Subjects:
1-cycle
14A15
14C25
14D06
14D10
14G40
Algebraic Geometry
avoidance lemma
Bertini-type theorem
hypersurface
Mathematics
moving lemma
Noether normalization
pictorsion
quasisection
rational equivalence
zero locus of a section
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subscheme C, and intersecting properly a closed set F. Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied if R is the ring of integers of a number field, or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1-cycles on a regular scheme X quasi-projective and flat over S. We also show the existence of a finite surjective S-morphism to the projective space P_S^d for any scheme X projective over S when X/S has all its fibers of a fixed dimension d.
ISSN:0012-7094
1547-7398
DOI:10.1215/00127094-2877293