Hypersurfaces in projective schemes and a moving lemma

TL;DR

This paper develops a technique to find hypersurfaces and finite quasi-sections in projective schemes over a base, with applications including a moving lemma for horizontal 1-cycles and embeddings into projective space.

Contribution

It introduces a new method for constructing hypersurfaces and quasi-sections in projective schemes, extending to cases with specific base conditions and providing a moving lemma for certain cycles.

Findings

Existence of hypersurfaces containing a given subscheme and intersecting a set properly.

Proof of a moving lemma for horizontal 1-cycles on regular schemes.

Construction of finite surjective morphisms to projective space for schemes over certain bases.

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 -> 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.