Open surfaces of small volume

TL;DR

This paper constructs new algebraic surfaces with minimal volume and canonical class, significantly improving known records for surfaces with specific singularities and divisors.

Contribution

It introduces explicit examples of surfaces with log terminal singularities and ample canonical class, achieving record low volumes and new configurations.

Findings

Constructed a surface with $K_X^2=1/48983$

Created a log canonical pair with $(K_X+B)^2=1/462$

Both examples set new records for minimal volumes

Abstract

We construct a surface with log terminal singularities and ample canonical class that has $K_X^2=1/48 983$ and a log canonical pair $(X,B)$ with a nonempty reduced divisor $B$ and ample $K_X+B$ that has $(K_X+B)^2 = 1/462$. Both examples significantly improve known records.