Smooth quotients of bi-elliptic surfaces

TL;DR

This paper investigates quotients of bi-elliptic surfaces by finite automorphisms, showing smooth quotients are either bi-elliptic or ruled surfaces, and concludes bi-elliptic surfaces lack Galois embeddings.

Contribution

It classifies smooth quotients of bi-elliptic surfaces and establishes that they cannot serve as Galois coverings of the projective plane.

Findings

Smooth quotients are either bi-elliptic or ruled surfaces with irregularity one.

Bi-elliptic surfaces cannot be Galois coverings of the projective plane.

Bi-elliptic surfaces do not admit Galois embeddings.

Abstract

We consider the quotient X of bi-elliptic surface by a finite automorphism group. If X is smooth, then it is a bi-elliptic surface or ruled surface with irregularity one. As a corollary any bi-elliptic surface cannot be Galois covering of projective plane, hence does not have any Galois embedding.