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 .
"Hypersurfaces in projective schemes and a moving lemma." Duke Math. J. 164 (7) 1187 - 1270, 15 May 2015. https://doi.org/10.1215/00127094-2877293