Let
F
|
K
F|K
be a function field in one variable and
V
\mathcal V
be a family of independent valuations of the constant field
K
.
K.
Given
v
∈
V
,
v\in \mathcal V ,
a valuation prolongation
v
\mathrm v
to
F
F
is called a constant reduction if the residue fields
F
v
|
K
v
F\mathrm v |Kv
again form a function field of one variable. Suppose
t
∈
F
t\in F
is a non-constant function, and for each
v
∈
V
v\in \mathcal V
let
V
t
V_{t}
be the set of all prolongations of the Gauß valuation
v
t
v_{t}
on
K
(
t
)
K(t)
to
F
.
F.
The union of the sets
V
t
V_{t}
over all
v
∈
V
v\in \mathcal V
is denoted by
V
t
.
\mathbfit {V}_{t}.
The aim of this paper is to study families of constant reductions
V
\mathbfit {V}
of
F
F
prolonging the valuations of
V
\mathcal V
and the criterion for them to be principal, that is to be sets of the type
V
t
.
\mathbfit {V}_{t}.
The main result we prove is that if either
V
\mathcal V
is finite and each
v
∈
V
v\in \mathcal V
has rational rank one and residue field algebraic over a finite field, or if
V
\mathcal V
is any set of non-archimedean valuations of a global field
K
K
satisfying the strong approximation property, then each geometric family of constant reductions
V
\mathbfit {V}
prolonging
V
\mathcal V
is principal. We also relate this result to the Skolem property for the existence of
V
\mathcal V
-integral points on varieties over
K
,
K,
and Rumely’s existence theorem. As an application we give a birational characterization of arithmetic surfaces
X
/
S
\mathcal X /S
in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to
P
S
1
.
\mathbb P ^{1}_{S}.