Abstract for the paper

On curves over valuation rings and morphisms to \(\textbf{P}^1\)

by Barry Green, Stellenbosch

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.