A remark on automorphisms of Kummer surfaces in characteristic p
The central object of this paper, the Kummer surface S associated with an algebraic curve C of genus \(2,\) was studied in great detail in the last century: If \(\iota\) is the canonical involution in the Jacobian Jac(C) and \(S':=Jac/<1,\iota >\), then S' has \(16\quad ordinary\) double points, and S is the minimal resolution of S'. The present author shows that there exists an elliptic fibration \(\pi: S\to {\mathbb{P}}^ 1_ k\) provided that char(k)\(\neq 2\). Six of the 32 distinguished lines on S provide sections for \(\pi\). If S is considered as an elliptic curve E over the function field K of \({\mathbb{P}}^ 1_ k\) one of these sections serves as unit element of E whereas the other five sections are shown to be K-rational points of infinite order. Here Kodaira's classification of singular fibres of an elliptic fibration plays the key role. This result is used to construct certain compact complex threefolds that are not Kähler.
scientific article; zbMATH DE number 3989545