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 -> 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.Comment: 64 page

Complete intersections: moduli, Torelli, and good reduction

“…Therefore any K-rational point of its generic fibre extends to a section over B. This follows from Zariski's main theorem, but also the much stronger statement proven in[33, Prop. 6.2].…”

Arithmetic hyperbolicity: automorphisms and persistence

“…Proof Choose a smooth Z-finitely generated subring A ⊂ k and an abelian scheme G → Spec A such that G k ∼ = G and such that G(Frac(A)) is Zariski dense in G; such data exists by Corollary 3.7. To conclude the proof, note that the geometric fibres of G → S contain no rational curves, so that G(A) = G(Frac(A)) by [16,Proposition 6.2].…”

Global Weierstrass equations of hyperelliptic curves

“…with an isomorphism σ. Using the structure of Pic(P(G)) (see for instance [2], Proposition 8.4(a)), there exists M ∈ Pic(T ) such that σ * O P(F ) (1) = O P(G) (1) ⊗ p * M . Taking p * and using the projection formula ([5], Proposition 5.2.32), we find…”