Log canonical thresholds of Burniat surfaces with $K^2 = 6$

Let $S$ be a Burniat surface with $K^2_S = 6$. Then we show that $\operatorname{glct}(S, K_S) = \frac{1}{2}$ by showing that $\operatorname{glct}(S, 2K_S) = \operatorname{lct} (S, E) = \frac{1}{4}$ for some divisor $E \in \lvert 2K_S \rvert$. This implies that Tian’s conjecture (which fails in general) holds for the polarized pair $(S, 2K_S)$, since the corresponding graded algebra is generated by sections of $H^0 (S, 2K_S)$.

Moreover we verify that any divisor $D \in \lvert mK_S \rvert$ such that $\operatorname{glct}(S, K_S) = \operatorname{lct}(S, \frac{1}{m} D)$ for a positive even integer $m$ is invariant under the $\mathbb{Z}^2_2$-action associated to the bicanonical map of $S$.

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Received 18 November 2018

Accepted 28 February 2019

Published 14 December 2020