Log Canonical Thresholds on Burniat Surfaces with $K^2 = 6$ via Pluricanonical Divisors

Abstract

Let $S$ be a Burniat surface with $K_S^2 = 6$ and $\varphi$ be the bicanonical map of $S$. In this paper we show optimal lower bounds of log canonical thresholds of members of pluricanonical sublinear systems of $S$ via Klein group $G$ induced by $\varphi$. Indeed, for a positive even integer $m$, the log canonical threshold of members of an invariant (resp. anti-invariant) part of $|mK_S|$ is greater than or equal to $1/(2m)$ (resp. $1/(2m-2)$). For a positive odd integer $m$, the log canonical threshold of members of an invariant (resp. anti-invariant) part of $|mK_S|$ is greater than or equal to $1/(2m-5)$ (resp. $1/(2m)$). The inequalities are all optimal.