Remark on complements on surfaces

TL;DR

This paper provides explicit bounds and characterizations of singularities and invariants for surfaces, including the first known explicit upper bounds for certain algebraic invariants like complements and Tian's alpha invariant.

Contribution

It offers the first explicit bounds for complements and Tian's alpha invariant on surfaces, and characterizes singularities of exceptional pairs in all dimensions.

Findings

Any exceptional Fano surface is rac{1}{42}-lc.

Any alabi-Yau surface has an n-complement with n pprox 10^{10^{10.5}}.

Tian's alpha invariant for any surface is pprox 10^{10^{10.2}}.

Abstract

We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is $\frac{1}{42}$-lc. As corollaries, we show that any $\mathbb R$-complementary surface $X$ has an $n$-complement for some integer $n\leq 192\cdot 84^{128\cdot 42^5}\approx 10^{10^{10.5}}$, and Tian's alpha invariant for any surface is $\leq 3\sqrt{2}\cdot 84^{64\cdot 42^5}\approx 10^{10^{10.2}}$. Although the latter two values are expected to be far from being optimal, they are the first explicit upper bounds of these two algebraic invariants for surfaces.