On curves over valuation rings and morphisms to \(\mathbb{P}^ 1\)
Given a valued field \((K,\nu)\) with valuation ring \(O_\nu\), an \(O_\nu\)-curve is a proper integral normal scheme \(X\) over \(O_\nu\) such that every irreducible component of every fibre has dimension 1. If \(K\) is relatively algebraically closed in \(F\) and \(F\) is a function field of transcendence degree 1 over \(K\) then the \(O_\nu\)-curves with function field \(F\) are classified. The classification uses valuation theoretic data. There are certain finite sets of valuations of \(F\) extending \(\nu\) that are said to be principal with constant reduction. Each such set \(V\) can be used to construct a canonical projective curve \(C_V\) over \(O_\nu\). Conversely, if \(X\) is an \(O_\nu\)-curve with function field \(F\) and if \(K \subseteq F\) satisfies a condition called geometric henselian then \(X\) determines a finite principal set \(V\) with constant reduction and \(X\) is isomorphic to the canonical curve \(C_V\). The extension \(K \subseteq F\) is geometric henselian, for example, if either \((K,\nu)\) is either henselian, or if the value group is a subgroup of \(\mathbb{Q}\) and the residue field is algebraic over its prime field.
scientific article; zbMATH DE number 935006