Translator Disclaimer
15 May 2015 Hypersurfaces in projective schemes and a moving lemma
Ofer Gabber, Qing Liu, Dino Lorenzini
Duke Math. J. 164(7): 1187-1270 (15 May 2015). DOI: 10.1215/00127094-2877293


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 P S d for any scheme X projective over S when X / S has all its fibers of a fixed dimension d .


Download Citation

Ofer Gabber. Qing Liu. Dino Lorenzini. "Hypersurfaces in projective schemes and a moving lemma." Duke Math. J. 164 (7) 1187 - 1270, 15 May 2015.


Received: 16 July 2011; Revised: 9 July 2014; Published: 15 May 2015
First available in Project Euclid: 14 May 2015

zbMATH: 06455741
MathSciNet: MR3347315
Digital Object Identifier: 10.1215/00127094-2877293

Primary: 14A15 , 14C25 , 14D06 , 14D10 , 14G40

Keywords: $1$-cycle , avoidance lemma , Bertini-type theorem , Hypersurface , moving lemma , Noether normalization , pictorsion , quasisection , rational equivalence , zero locus of a section

Rights: Copyright © 2015 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.164 • No. 7 • 15 May 2015
Back to Top