Unirational elliptic surfaces in characteristic 2
A unirational elliptic surface \(f:X\to\mathbb{P}^ 1\) defined over an algebraically closed field \(k\) of characteristic \(p\) is of base change type if there exists a surjective morphism \(g:C\to\mathbb{P}^ 1\) of curves such that \(X\times_{\mathbb{P}^ 1}C\) is a rational surface over \(k\). The author classifies completely the minimal Weierstrass normal forms of irrational unirational elliptic surfaces of base change type with sections in case \(k\) has characteristic 2; there are 21 types of them. In case char \(k>2\), \textit{T. Katsura} [Tôhoku Math. J., II. Ser. 33, 521- 553 (1981; Zbl 0466.14008)] classified such surfaces. -- As a by-product, the author shows that an irrational unirational elliptic surface of base change type with a section as a supersingular K3 surface, i.e., \(\rho(X)=b_ 2(X)\), if char \(k=2\), while this is not always the case if char \(k>2\).
scientific article; zbMATH DE number 218423