Let ${\cal O}_v$ be a valuation ring with valuation $v$ and quotient field $K.$ The aim of this paper is to study proper, integral, normal ${\cal O}_v$-curves $\cal X,$ ($=\,{\cal O}_v$-schemes of pure relative dimension 1), and more generally curves defined over a normal, integral scheme $S$, whose local rings at the closed points are valuation rings. The central result gives a precise characterization of such ${\cal O}_v\!$-curves as a normalisation of ${\bf P}^1_{{\cal O}_v}$ in the function field $\kappa({\cal X}),$ for the class of valuation rings ${\cal O}_v$ which satisfy the {\it Local Skolem Property.\/} The Local Skolem Property at $v$ is a criterion for the solvability of systems of algebraic diophantine equations in rings of algebraic $v\!$-integers. This class includes all {\it valuation rings whose value groups have rational rank 1and whose residue fields are algebraic over a finite field,\/} so in particular the {\it global fields equipped with non-archimedian valuations.\/} The {\it henselian valuation rings,\/} irrespective of their value group and residue field, also belong to this class.

Last update: February 4, 1999