## 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

### MSC:

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