Unirational quasi-elliptic surfaces in characteristic 3. (English) Zbl 0358.14019

A non-singular projective surface is called “quasi-elliptic”; if it admits a morphism \(f\) to a curve \(C\), where almost all the fibres of \(f\) are irreducible singular rational curves of arithmetic genus one. By a theorem of Tate such a surface can exist only in the rase where the characteristic of the groundfieldis 2 or 3. The author considers the rase of a quasi-elliptic surface \(X\) of characteristic 3 for which \(f\) admits a rational section, and \(X\) is unirational (equivalently: \(C\) is rational). He computes some of the basic invariants of there surfaces, under further hypotheses.
Reviewer: Barry Mazur


14J15 Moduli, classification: analytic theory; relations with modular forms
14M20 Rational and unirational varieties
14G15 Finite ground fields in algebraic geometry