Remark on complements on surfaces
Forum of Mathematics, Sigma, Tome 11 (2023)
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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.
@article{10_1017_fms_2023_35,
author = {Jihao Liu},
title = {Remark on complements on surfaces},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.35},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.35/}
}
Jihao Liu. Remark on complements on surfaces. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.35
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