Let $F|K$ be a function field in 1 variable and $\bf V$ be a family of independent valuations of the constant field $K.$ Given $v\in\bf V,$ a valuation prolongation $\rm v$ to $F$ is called a {\it constant reduction\/} if the residue fields $F{\rm v}|Kv$ again form a function field of one variable. Suppose $t\in F$ is a non-constant function and for each $v\in\bf V$ let $V_t$ be the set of all prolongations of the Gau{\ss} valuation $v_t$ on $K(t)$ to $F.$ The union of the sets $V_t$ over all $v\in\VV$ is denoted by ${\cal V}_{\!t}.$

The aim of this paper is to study families of constant reductions $\cal V$ of $F$ prolonging the valuations of $\bf V$ and the criterion that they are principal, that is that they are sets of the type ${\cal V}_{\!t}.$ The main result we prove is that if either, $\bf V$ is finite and each $v\in\bf V$ has rational rank 1 and residue field algebraic over a finite field, or if $\bf V$ is any set of non-archimedean valuations of a global field $K$ satisfying the strong approximation property, then each geometric family of constant reductions $\cal V$ prolonging $\bf V$ is principal. We also relate this result to the {\it Skolem property\/} for the existence of $\bf V\!$-integral points on varieties over $K,$ and Rumely's existence theorem. As an application we give a {\it birational characterization\/} of arithmetic surfaces ${\cal X}/S$ in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to ${\bf P}^1_S.$

Last update: February 4, 1999