Surfaces with $K^2=2\mathcal{X}-2$ and $p_g\geq 5$

TL;DR

This paper classifies minimal surfaces of general type with specific invariants, focusing on their canonical maps and fibrations, revealing new structural insights for surfaces with high geometric genus.

Contribution

It provides a complete characterization of minimal surfaces with $p_g \,\geq 5$ and $K^2=2p_g$, especially detailing their canonical maps and genus 2 fibrations.

Findings

For $p_g\geq 8$, the canonical map is generically finite of degree 2.

For $p_g\geq 13$, surfaces have a unique genus 2 fibration.

For $p_g\leq 12$, there are two additional classes with non-birational canonical maps.

Abstract

This note describes minimal surfaces $S$ of general type satisfying $p_g\geq 5$ and $K^2=2p_g$. For $p_g\geq 8$ the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of such surfaces with non birational canonical map. It turns out that if $p_g\geq 13$, $S$ has always an (unique) genus 2 fibration, whose non 2-connected fibres can be characterized, whilst for $p_g\leq 12$ there are two other classes of such surfaces with non birational canonical map.