On the unirationality of del Pezzo surfaces of degree 2
type: Item
instance of: scholarly article
title: On the unirationality of del Pezzo surfaces of degree 2
zbMATH Open document ID: 1319.14016
author: Cecília Salgado
author: Damiano Testa
author: Anthony Várilly-Alvarado
publication date: 2014-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.6798
Mathematics Subject Classification ID: 14E08
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 6327930
zbMATH Keywords: del Pezzo surface
zbMATH Keywords: unirationality
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W2109157247
arXiv ID: 1304.6798
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cites work: Cox rings of degree one del Pezzo surfaces
cites work: Q3237900
DOI: 10.1112/JLMS/JDU014
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published in: Journal of the London Mathematical Society. Second Series
review text: It was shown by \textit{V. A. Iskovskikh} [Math. USSR, Izv. 14, 17--39 (1980; Zbl 0427.14011)] that any smooth projective geometrically rational (i.e., rational over \(\bar k\)) surface \(X\) over a field \(k\), containing a rational point, is birational over \(k\) to either a del Pezzo surface or a conic bundle. It is also known that del Pezzo surfaces of degree \(\geq 3\) are always unirational (over~\(k\)), and \textit{Yu. I. Manin} [Cubic forms. Algebra, geometry, arithmetic. Translated from Russian by M. Hazewinkel. London: North- Holland Publishing Company; New York: American Elsevier Publishing Company, Inc (1974; Zbl 0277.14014)] showed the same for many degree~\(2\) surfaces.NEWLINENEWLINEIn this paper the authors almost complete Manin's work. More precisely, they show that if \(k\) is a finite field and \(X\) is a degree~\(2\) del Pezzo surface then \(X\) is unirational, with three possible exceptions (two over \({\mathbb F}_3\) and one over \({\mathbb F}_9\)). In the meantime, these exceptional cases have also been shown to be unirational by Festi and van~Luijk.NEWLINENEWLINEThe proof follows Manin's in detail. The idea behind Manin's proof is that through a rational point not contained in some divisor there is a rational curve: this curve has many rational points and there is a (different) rational curve through each of them, and this family uniformises \(X\). The difficulty, then, is to produce rational curves, with the possibility that all the rational points on \(X\) are in the forbidden divisor.NEWLINENEWLINEThere are several different cases. The main one is that if there are eight distinct points which are not contained in the branch locus of the anticanonical morphism \(X\to{\mathbb P}^2\), then there is a rational curve through one of the points. The case of characteristic~\(2\) requires special treatment. Along the way, the authors correct the definition of the divisor that must be avoided in Manin's argument (the correction is important for them…
reviewed by: G. K. Sankaran
description: scientific article; zbMATH DE number 6327930