An Enriques surface without a rational point over \(\mathbb C((t))\)
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 \(\mathbb{C}(t)\).
Enriques surface without a rational point
scientific article; zbMATH DE number 2057058