Log canonical thresholds on Burniat surfaces with \(K^2 = 6\) via pluricanonical divisors (Q2102526)

A Burniat Surface \(S\) is a minimal surface of general type with \(p_g(S)=q(S)=0\) and \(K_S^2=2,3,4,5,6\). The so-called Primary Burniat surfaces are the ones with \(K^2=6\). Given \(D\) an effective Cartier divisor on a variety \(X\) the log canonical threshold of the pair \((X,D)\) is defined by \[ \mathrm{lct}(X,D):= \sup \{c\in \mathbb{Q} \mid (X,cD) \text{ is a log canonical pair}\}, \] and the global log canonical threshold for a pair \((X,L)\) with \(L\) an ample divisor on \(X\) is defined as \[ \mathrm{glct}(X,L):=\inf \{lct(X,D)\mid D \text{ is an effective }\mathbb{Q}\text{-divisor linearly } \mathbb{Q}\text{-equivalent to } L \}. \] The authors in the paper for Burniat primary surfaces \(S\) found an optimal lower bound of the log canonical threshold of some members of the pluri-canonical system \(|mK_S|\) using the action induced by the bi-canonical cover of the Klein group \(\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\). In some way, it is an extension of a previous work done by the authors where he proves that the global log canonical threshold of a primary Burniat surface is \(\frac{1}{2}\). See, [\textit{I.-K. Kim} and \textit{Y. Shin}, Math. Res. Lett. 27, No. 4, 1079--1094 (2020; Zbl 1457.14088)]