Remark on complements on surfaces

We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is 1/42 -lc. As corollaries, we show that any R -complementary surface X has an n -complement for some integer n≤ 192· 84128· 42⁵≈ 10^1010.5 , and Tian’s alpha invariant for any surface is ≤ 3√2· 8464· 42⁵≈ 10^1010.2 . 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.