Hypersurfaces in projective schemes and a moving lemma. (English) Zbl 1432.14002

Gabber, Ofer; Liu, Qing; Lorenzini, Dino

Duke Math. J. 164, No. 7, 1187-1270 (2015).

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 \(Z\to S\), \(\mathrm{Pic}(Z)\) 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 \(\mathbb{P}_S^d\) for any scheme \(X\) projective over \(S\) when \(X/S\) has all its fibers of a fixed dimension \(d\).

MSC:

14A15	Schemes and morphisms
14C25	Algebraic cycles
14D06	Fibrations, degenerations in algebraic geometry
14D10	Arithmetic ground fields (finite, local, global) and families or fibrations
14G40	Arithmetic varieties and schemes; Arakelov theory; heights

Keywords:

avoidance lemma; Bertini-type theorem; hypersurface; moving lemma; Noether normalization; \(1\)-cycle; pictorsion; quasisection; rational equivalence; zero locus of a section

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