Every rational surface is separably split
type: Item
instance of: scholarly article
title: Every rational surface is separably split
zbMATH Open document ID: 0683.14014
DOI: 10.1007/BF02566770
author: Kevin R. Coombes
published in: Commentarii Mathematici Helvetici
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140125
review text: A smooth projective surface defined over a field F is called rational over F if X is birational to \({\mathbb{P}}^ 2\) over the algebraic closure of F. An extension E/F is said to split X if \(X_ E=X\otimes_ FE\) is birational with \({\mathbb{P}}^ 2\) by a sequence of monoidal transformations centered at E-points. - The main theorem of this paper asserts that every rational surface admits a splitting field E/F which is a separable extension. The principal tool of the proof is Iskovskih's classification of minimal rational surfaces, for which the following ``adjunction lemma'' is needed: if X is a minimal surface over a separably closed field with \(q=P_ 2=0\), then for any divisor D on X there exists an integer \(n_ 0\) such that \(| D+nK| =\emptyset\) for \(n\geq n_ 0\). This is proved in {\S} 1. So far the lemma had been proved for perfect ground field only. There are some misprints, which are easy to correct.
reviewed by: H.Matsumura
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 4118499
zbMATH Keywords: every rational surface admits a splitting field
zbMATH Keywords: adjunction lemma
zbMATH Keywords: divisor
Wikidata QID: Q57556835
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4118499