Differential function fields and moduli of algebraic varieties
type: Item
instance of: scholarly article
title: Differential function fields and moduli of algebraic varieties
zbMATH Open document ID: 0613.12018
published in: Lecture Notes in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The aim of this monograph is to establish a link between the theory of algebraic differential equations (ADE) with no movable singularity and the Galois theory of ADE's. In the one-dimensional case a modern treatment of the first mentioned theory was given by \textit{M. Matsuda} [First order algebraic differential equations (Lect. Notes Math. 804) (1980; Zbl 0447.12014)], while the second mentioned theory was given its final form in papers of Kolchin [e.g. \textit{E. R. Kolchin}, ''Differential algebra and algebraic groups'' (1973; Zbl 0264.12102)]. After giving same background in chapter I, in chapter II the author generalizes most of Matsuda's results to the higher dimensional case. Let K be a field of characteristic zero, \(\Delta\) a - not necessarily finite - set of differential operators of K, \(F| K\) a function field. If \(\Delta\) is finite then \(F| K\) has a \(\Delta\)-model; in general a \(\Delta\)-function field in said to have no movable singularity if it has a projective \(\Delta\)-model. \(F| K\) is called split if \(F=K(F^{\Delta})\). Such function fields have no movable singularity. The two important resuls of chapter II are the following: Let K be an algebraically closed \(\Delta\)-field and V a projective variety over K. Then \(K^{\Delta}\) is a field of definition of V (theorem 1.1). From Kolchin's theory the author uses the concepts of weak and strong normality for \(\Delta\)-fields \(F| K:\) \(F| K\) is called weakly normal if \(F^{Gal_{\Delta}(F| K)}=K\) and strongly normal if in addition for any K-isomorphism \(\sigma\) of F into a \(\Delta\)-extension E of F both extensions \(F\cdot \sigma F/F\) and \(F\cdot \sigma F/\sigma F\) are split. Using theorem 1.1 the author shows (theorem 2.1 and corollary 2.2): If \(F| K\) has no movable singularity then there exists a \(\Delta\)-field extension \(K_ 1| K\) with \(K_ 1^{\Delta}| K^{\Delta}\) algebraic such that \(Q(F\otimes_ K K_ 1)/K_ 1\) is split. If moreover \(\Delta\) is finite and the operat…
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 12-02
zbMATH DE Number: 3991636
zbMATH Keywords: differential function field
zbMATH Keywords: algebraic differential equations
zbMATH Keywords: ADE
zbMATH Keywords: Galois theory
zbMATH Keywords: no movable singularity
zbMATH Keywords: Picard-Vessiot extension
zbMATH Keywords: Kolchin's theory
zbMATH Keywords: normal function fields
zbMATH Keywords: linearization property
zbMATH Keywords: complete integrable Hamiltonian systems
zbMATH Keywords: Euler equations
author: Alexandru Buium
reviewed by: Karl-Heinz Kiyek
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3991636