A note on a theorem of Xiao Gang

TL;DR

This paper provides a new proof of Xiao Gang's 1985 theorem on the bicanonical system of complex surfaces of general type with geometric genus zero, extending understanding of these surfaces' properties.

Contribution

It offers a novel proof of Xiao Gang's theorem specifically for surfaces with geometric genus zero, filling a gap in the existing literature.

Findings

The bicanonical system of surfaces with p_g=0 is not composed of a pencil.

The proof confirms the theorem for a previously unaddressed class of surfaces.

Supports the broader understanding of the geometry of complex surfaces.

Abstract

In 1985 Xiao Gang proved that the bicanonical system of a complex surface $S$ of general type with $p_2(S)>2$ is not composed of a pencil [Bull. Soc. Math. France, 113 (1985), 23--51]. When in the end of the 80's it was finally proven that $| 2K_S|$ is base point free, whenever $p_g\geq 1$, the part of this theorem concerning surfaces with $p_g\geq 1$ became trivial. In this note a new proof of this theorem for surfaces with $p_g=0$ is presented.