GABBER, Ofer; LIU, Qing; LORENZINI, Dino

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GABBER, Ofer
Institut des Hautes Études Scientifiques [IHES]

LIU, Qing
Équipe Théorie des Nombres

LORENZINI, Dino

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Duke Mathematical Journal. 2015, vol. 164, n° 7, p. 1187-1270

Duke University Press

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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.< Leer menos

Palabras clave en inglés

Noether normalization

Avoidance lemma

Bertini-type theorem

Hypersurface

Moving lemma

Multisection

1-cycle

Pictorsion

Quasi-section

Rational equivalence

Zero locus of a section

Noether normalization.

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Hypersurfaces in projective schemes and a moving lemma

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