The minimal volume of surfaces of log general type with non-empty non-klt locus
Jihao Liu, Wenfei Liu
TL;DR
The authors address the problem of locating the minimal volume among projective log canonical surfaces of general type with an ample canonical divisor, focusing on those with a non-empty non-klt locus. By combining nef/pseudo-effective threshold gaps (notably from LS23), a detailed study of lc surface singularities, and a construction anchored in K3 symmetry (order-11 non-symplectic automorphisms), they identify the exact minimal nklt volume as $\frac{1}{825}$ and construct a unique ample model $V$ realizing this bound. They show $V$ embeds canonically as a degree $86$ hypersurface in ${\mathbb{P}}(6,11,25,43)$ and admits a one-parameter deformation to klt stable hypersurfaces, yielding a complete rational curve in the moduli space $M_{\frac{1}{825}}$; this also establishes $\frac{1}{825}$ as the smallest accumulation point of volumes for projective lc surfaces. The work further clarifies the landscape of stable surface volumes below $\frac{1}{825}$, presents a precise minimal example, and discusses implications for moduli and accumulation phenomena in stable pairs.
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 $\textit{complete}$ 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 $\frac{1}{825}$.