Extremal rational elliptic surfaces in characteristic \(p\). II: Surfaces with three or fewer singular fibres
In this part II, we complete the classification of extremal rational elliptic surfaces in characteristic \(p\), begun in part I of this paper [Math. Z. 207, No. 3, 429-438 (1991; Zbl 0757.14012)]. \textit{R. Miranda} and \textit{U. Persson} [Math. Z. 193, 537-558 (1986; Zbl 0652.14003)] classified all rational elliptic surfaces over the complex numbers such that the Mordell-Weil group of the generic fibre is finite. They called these surfaces extremal rational elliptic surfaces. They found 16 families of such surfaces. All but one of these families have only one member, and the exceptional family depends on one parameter. --- In part I of this paper on extremal rational elliptic surfaces in characteristic \(p\), we classified those where the singular fibres are semi-stable. These are the characteristic \(p\) analogues of the surfaces studied by \textit{A. Beauville} [in: C. R. Acad. Sci., Paris, Sér. I 294, 657-660 (1982; Zbl 0504.14016)], and we called them Beauville surfaces. In this paper, we classify all other extremal rational elliptic surfaces. The classification is identical to the classification in characteristic zero in all characteristics except two and three. (There is one exceptional case in characteristic five.) The classification in characteristics two and three looks quite different. This is due to the presence of a wild ramification term in the formula of Néron-Ogg-Shararevich, which appears only in these characteristics.
classification of extremal rational elliptic surfaces in characteristic \(p\)
scientific article; zbMATH DE number 750857