Every rational surface is separably split
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.
every rational surface admits a splitting field
scientific article; zbMATH DE number 4118499