Combinatorially minimal Mori dream surfaces of general type

TL;DR

This paper introduces a new approach using Cox rings and combinatorially minimal Mori dream spaces to study minimal surfaces of general type with geometric genus zero, providing new insights and classifications.

Contribution

It develops the theory of combinatorially minimal Mori dream surfaces and applies it to analyze minimal surfaces of general type with p_g=0, including their fibrations and singularities.

Findings

Explicit examples of Mori dream surfaces with p_g=0 and their effective cones

Analysis of fibrations and singularities of minimal models

Many minimal surfaces of general type with p_g=0 arise from these Mori dream surfaces

Abstract

In this paper, we suggest a new approach to study minimal surfaces of general type with $p_g=0$ via their Cox rings, especially using the notion of combinatorially minimal Mori dream space introduced by Hausen. First, we study general properties of combinatorially minimal Mori dream surfaces. Then we discuss how to apply these ideas to the study of minimal surfaces of general type with $p_g=0$ which are very important but still mysterious objects. In our previous paper, we provided several examples of Mori dream surfaces of general type with $p_g=0$ and computed their effective cones explicitly. In this paper, we study their fibrations, explicit combinatorially minimal models and discuss singularities of the combinatorially minimal models. We also show that many minimal surfaces of general type with $p_g=0$ arise from the minimal resolutions of combinatorially minimal Mori dream surfaces.