Hypersurfaces in projective schemes and a moving lemma

Abstract

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 $Z\to S$, $Pic\left(Z\right)$ 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 $d$.