On the unirationality of del Pezzo surfaces of degree 2
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, but not for Manin's original result).
scientific article; zbMATH DE number 6327930
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