A Characterization of Inoue Surfaces with p_g=0 and K²=7

2017/08/27 by Yifan Chen, Chen, Yifan, YongJoo Shin +1
Computer Science · Mathematics · #14J10 #14J29 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1708.08061

openalex publication_date 2017/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Inoue constructed the first examples of smooth minimal complex surfaces of general type with p_g=0 and K²=7.These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group ℤ₂× ℤ₂. For such a surface S, the bicanonical map of S has degree 2 and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components:one is a genus 3 curve with self-intersection number 0 and the other is a genus 2 curve with self-intersection number -1. Conversely, assume that S is a smooth minimal complex surface of general type with p_g=0, K²=7 and having an involution σ. We show that, if the divisorial part of the fixed locus of σ consists of two irreducible components R₁ and R₂,with g(R₁)=3, R₁²=0, g(R₂)=2 and R₂²=-1, then the Klein group ℤ₂× ℤ₂ acts faithfully on S and S is indeed an Inoue surface.

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