The minimal volume of surfaces of log general type with non-empty non-klt locus

2023/08/28 by Jihao Liu, Liu, Jihao, Wenfei Liu +1 · 1 citation
Mathematics · #14B05 #14E30 #14J29 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2308.14268

openalex publication_date 2023/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is (1)/(825). Furthermore, the ample model V achieving the minimal volume is determined uniquely up to isomorphism. The canonical embedding presents V as a degree 86 hypersurface of \mathbb P(6,11,25,43). This motivates a one-parameter deformation of V to klt stable surfaces within the weighted projective space. Consequently, we identify a complete rational curve in the corresponding moduli space M_(1)/(825). As an important application, we deduce that the smallest accumulation point of the set of volumes for projective log canonical surfaces equals (1)/(825).

Cited by

• On local accumulation complexity of the set of log canonical volumes in dimension ≥ 2

Related

• The minimal volume of log surfaces of general type with positive geometric genus
• Remarks on the minimal model theory for log surfaces in the analytic setting
• Open surfaces of small volume
• Almost minimal models of log surfaces
• On minimal model theory for algebraic log surfaces