Reduction of Algebraic Varieties with Respect to a Discrete Valuation of the Basic Field

Abstract

In the arithmetic theory of algebraic varieties it is useful to have a general theory on modulo of varieties. Here means, roughly spoken, the following process: let V be an algebraic variety defined over a field k with a discrete valuation, and denote by p the maximal ideal of that valuation; then we have an algebraic variety VP over the residue field K of t defined by the equations for v modulo p; such a variety V may be called the variety obtained from the variety V by the reduction modulo p. When one tries to construct the general theory of reduction in this sense, the first thing to do will be to investigate how the algebraico-geometric properties of V behave in the reduction process. The object of this paper is concerned with that problem. We shall give a precise definition of the reduction of varieties and the reduction of cycles on a variety. In the case when the ambient varieties v and V are both absolutely irreducible, we shall show that the reduction of cycles on V defines a linear mapping from the group of rational 1-cycles over Ic into the group of rational V-cycles over K of the same dimension; and shall show that the operations on cycles (the intersection-product, the direct product and the algebraic projection) are all preserved by that mapping (Section 4). If we consider reduction as specialization, this result may be regarded as a specialization-theory of cycles on algebraic varieties, in a certain sense. In fact, from this reduction-theory of cycles we can easily obtain a more general specialization-theory of cycles (Section 5). The latter will be a generalization of the specialization-theory of cycles which has been given by T. Matsusaka [4] and P. Samuel [6]. Our theory may be also called a local theory of varieties at a divisor p, whereas 4 is not a usual divisor on varieties. We shall also show the following fact: let K be a field with a set of infinitely many valuations which satisfies a certain condition (condition (I) in Section 6); if v is an absolutely irreducible variety defined over K, then for almost all those valuations, it remains absolutely irreducible by the reduction process.