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

TL;DR

This paper determines the minimal volume of certain algebraic surfaces with specific singularities, identifies the unique surface achieving this volume, and explores its moduli space, revealing a rational curve and the smallest accumulation point of volumes.

Contribution

It establishes the exact minimal volume for surfaces of log general type with non-klt locus and characterizes the unique minimal surface, linking it to moduli space geometry.

Findings

Minimal volume is 1/825 for the specified surfaces.

The minimal surface is uniquely determined up to isomorphism.

A rational curve in the moduli space is constructed, and the smallest accumulation point of volumes is identified.

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 $\frac{1}{825}$.