2014/04/22 by Ofer Gabber, Qing Liu, Dino Lorenzini
Mathematics · #Advanced Algebra and Geometry #Affine transformation #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Dimension (graph theory) #Discrete mathematics #Lemma (botany) #Mathematics #Morphism #Pure mathematics #Surjective function #math.AG #msc:14A15 #msc:14C25 #msc:14D06 #msc:14D10 #msc:14G40
paper · pdf · doi:10.1215/00127094-2877293
published as Duke Math. J. 164, no. 7 (2015), 1187-1270 · 64 pages
arxiv created 2014/04/22 · openalex publication_date 2015/05/14 · arxiv updated 2016/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 P S d for any scheme X projective over S when X / S has all its fibers of a fixed dimension d .