The index of an algebraic variety

• 成果类型: Article
• 署名作者: Gabber, Ofer; Liu, Qing; Lorenzini, Dino
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); Inria; Universite de Bordeaux; University System of Georgia; University of Georgia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0418-z
• 发表日期: 2013
• 页码: 567-626
• 关键词: picard-groups; CURVES; schemes; domains; points

摘要

Let K be the field of fractions of a Henselian discrete valuation ring . Let X (K) /K be a smooth proper geometrically connected scheme admitting a regular model . We show that the index delta(X (K) /K) of X (K) /K can be explicitly computed using data pertaining only to the special fiber X (k) /k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an -scheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant gamma(A) of a singular local ring : the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all -primary ideals Q in . We relate this invariant gamma(A) to the index of the exceptional divisor in a resolution of the singularity of , and we give a new way of computing the index of a smooth subvariety X/K of over any field K, using the invariant gamma of the local ring at the vertex of a cone over X.