Surfaces of general type with geometric genus zero: a survey

TL;DR

This survey reviews recent progress in classifying surfaces of general type with geometric genus zero, providing an updated list of known surfaces for specific values of the self-intersection number of the canonical divisor.

Contribution

It offers a comprehensive overview of the current state of classification and new constructions of these surfaces, updating the existing catalog for K^2 from 1 to 7.

Findings

Updated list of surfaces with p_g=0 for K^2=1,...,7

Summary of new constructions and classification progress

Discussion of key aspects in the classification process

Abstract

In the last years there have been several new constructions of surfaces of general type with $p_g=0$, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with $p_g=0$, and gives an updated list of the existing surfaces, in the case where $K^2= 1,...,7$. It also focuses on certain important aspects of this classification.