An Enriques surface without a rational point over \(\mathbb C((t))\)
type: Item
instance of: scholarly article
title: An Enriques surface without a rational point over \(\mathbb C((t))\)
zbMATH Open document ID: 1040.14020
author: Guillaume Lafon
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-03-15T00:00:00Z
review text: Serre, in a letter to Grothendieck [see \textit{P. Colmez} and \textit{J. P. Serre} (eds.), Correspondance Grothendieck-Serre, Doc. Math. (Paris), Soc. Math. Fr. (2001; Zbl 0986.01019)], outlined the following conjecture: Let \(X\) be a nonsingular projective variety, geometrically connected, defined over the field \(\mathbb{C}(C)\), with \(C\) a complex algebraic curve and such that \(H^i(X, {\mathcal O}_X)=0 \) for \(i>0\). \(X\)\, always possesses a rational point. Recently \textit{T. Graber, J. Harris, B. Mazur}, and \textit{J. Starr} [http://arxiv.org/abs/math?0210225] established the falsehood of this conjecture, showing the existence of an Enriques surface \(X\) over \(\mathbb{C}(C)\) without a rational point. However in such a work neither the curve \(C\) neither the surface \(X\) are given explicitly. In the present paper the author provides an explicit example exhibiting an Enriques surface over \(\mathbb{C}(t)\) (that is to say with \(C\) the complex proyective line) without a rational point on \(\mathbb{C}((t))\) and therefore neither on \(\mathbb{C}(t)\).
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 2057058
zbMATH Keywords: Enriques surfaces
zbMATH Keywords: rationals points
zbMATH Keywords: Enriques surface without a rational point
reviewed by: Juan Tena Ayuso
MaRDI profile type: Publication
full work available at URL: j.crma.2003.10.031
OpenAlex ID: W2016510348
cites work: Q5749404
cites work: Q4175362
cites work: Q4888847
cites work: Correspondence Grothendieck-Serre
cites work: Rational connectivity and sections of families over curves
cites work: Lie algebras and Lie groups. 1964 lectures, given at Harvard University.
cites work: Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack
DOI: 10.1016/J.CRMA.2003.10.031
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description: scientific article; zbMATH DE number 2057058