On curves over valuation rings and morphisms to \(\mathbb{P}^ 1\)
type: Item
instance of: scholarly article
title: On curves over valuation rings and morphisms to \(\mathbb{P}^ 1\)
zbMATH Open document ID: 0873.14038
author: Barry W. Green
published in: Journal of Number Theory
publication date: 1997-10-30T00:00:00Z
review text: 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.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 14J99
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 14H99
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 11G99
zbMATH DE Number: 935006
zbMATH Keywords: geometric henselian extension
zbMATH Keywords: valued field
zbMATH Keywords: function field
zbMATH Keywords: canonical projective curve
MaRDI profile type: Publication
full work available at URL: jnth.1996.0098
OpenAlex ID: W2093083026
DOI: 10.1006/JNTH.1996.0098
description: scientific article; zbMATH DE number 935006