Duke Mathematical Journal
- Duke Math. J.
- Volume 164, Number 7 (2015), 1187-1270.
Hypersurfaces in projective schemes and a moving lemma
Let be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes of 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 containing a given closed subscheme and intersecting properly a closed set .
Assume now that the base is the spectrum of a ring such that for any finite morphism , is a torsion group. This condition is satisfied if 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 -cycles on a regular scheme quasi-projective and flat over . We also show the existence of a finite surjective -morphism to for any scheme projective over when has all its fibers of a fixed dimension .
Duke Math. J., Volume 164, Number 7 (2015), 1187-1270.
Received: 16 July 2011
Revised: 9 July 2014
First available in Project Euclid: 14 May 2015
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Primary: 14A15: Schemes and morphisms 14C25: Algebraic cycles 14D06: Fibrations, degenerations 14D10: Arithmetic ground fields (finite, local, global) 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Gabber, Ofer; Liu, Qing; Lorenzini, Dino. Hypersurfaces in projective schemes and a moving lemma. Duke Math. J. 164 (2015), no. 7, 1187--1270. doi:10.1215/00127094-2877293. https://projecteuclid.org/euclid.dmj/1431608068