Non-rationality of some fibrations associated to Klein surfaces

TL;DR

This paper investigates the rationality of polynomial fibrations associated with Klein surfaces derived from finite linear group quotients, revealing non-rationality in certain cases and describing their automorphism groups.

Contribution

It establishes the non-rationality of generic fibers for Klein surfaces of types D and E, and characterizes their automorphism groups as finite-dimensional linear groups.

Findings

Non-rationality of generic fibers in cases D and E.

Rationality of fibers in case A.

Automorphism groups are finite-dimensional linear groups in cases D and E.

Abstract

We study the polynomial fibration induced by the equation of the Klein surfaces obtained as quotient of finite linear groups of automorphisms of the plane; this surfaces are of type A, D, E, corresponding to their singularities. The generic fibre of the polynomial fibration is a surface defined over the function field of the line. We proved that it is not rational in cases D, E, although it is obviously rational in the case A. The group of automorphisms of the Klein surfaces is also described, and is linear and of finite dimension in cases D, E; this result being obviously false in case A.