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

Abstract

We show that the minimal volume of surfaces of log general type, with non-empty non-klt locus on the ample model, is $\frac{1}{825}$. 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 $\text{complete}$ rational curve in the corresponding moduli space $M_{\frac{1}{825}}$. As an important application, we deduce that the smallest accumulation point of the set of volumes for projective log canonical surfaces equals $\frac{1}{825}$.

24 pages; Remark 2.10 added to indicate the relation to stable surfaces with p_g=1 and minimal volume; the proof of theorem 3.4 slightly clarified; to appear in Algebraic Geometry