On accumulation points of volumes of log surfaces

TL;DR

This paper investigates the accumulation points of volumes of log canonical surfaces with coefficients in a set satisfying the descending chain condition, showing they can be realized with specific coefficient properties and solving a conjecture about their rationality.

Contribution

It characterizes accumulation points of volumes of log surfaces with coefficients in certain sets and proves all such points are rational when coefficients are rational, also providing bounds for standard coefficients.

Findings

Accumulation points can be realized with coefficients in the closure of the set.

All accumulation points are rational if the coefficient set is rational.

Bounds are established for the minimal accumulation point in the standard coefficients case.

Abstract

Let $\mathcal C\subset(0,1]$ be a set satisfying the descending chain condition. We show that any accumulation point of volumes of log canonical surfaces $(X, B)$ with coefficients in $\mathcal C$ can be realized as the volume of a log canonical surface with big and nef $K_X+B$ and coefficients in $\overline{\mathcal C}\cup\{1\}$, with at least one coefficient in $Acc(\mathcal C)\cup\{1\}$. As a corollary, if $\overline{\mathcal C}\subset\mathbb Q$ then all accumulation points of volumes are rational numbers, solving a conjecture of Blache. For the set of standard coefficients $\mathcal C_2=\{1-\frac{1}{n}\mid n\in\mathbb N\}\cup\{1\}$ we prove that the minimal accumulation point is between $\frac1{7^2\cdot 42^2}$ and $\frac1{42^2}$.