📝 SummarXiv

Abstract

A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this paper, we study the structure of bielliptic surfaces over a field of characteristic different from $2$ and $3$, in order to prove the Shafarevich conjecture for bielliptic surfaces with rational points. Furthermore, we demonstrate that the Shafarevich conjecture generally fails for bielliptic surfaces without rational points. In particular, this paper completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension $0$. In the second part of this paper, we study a N\'{e}ron model of a bielliptic surface. We establish the potential existence of a N\'{e}ron model for a bielliptic surface when the residual characteristic is not equal to $2$ or $3$.