On index one covers of two-dimensional purely log terminal singularities in positive characteristic (Q1762701)

Let \(k\) be an algebraically closed field of characteristic \(p\). Let \(S\) be a normal surface over \(k\). A divisor \(\Delta=\sum d_i D_i\) on \(S\) is called a standard \(\mathbb Q\)-boundary if \(d_i=1-\frac{1}{b_i}, b_i\in \mathbb N\cup \{\infty\}\). Let \(f:S'\to S\) be a proper birational morphism, \(S'\) normal, and write \[ K_{S'}+\Delta'=f^\ast(K_S+\Delta)+\sum_E a(E, \Delta) E, \] \(\Delta'\) the strict transform of \(\Delta, E\) the exceptional divisor. Let \(\text{discrep}(S, \Delta) = \inf_E\{a(E, \Delta) \mid E \text{ exceptional divisor}\}.\) \((S, \Delta)\) is called purely \(\log\) terminal (resp. canonical) if discrep \((S, \Delta) > -1\) (resp. \(\geq 0\)). The index \(r\) of \((S, \Delta)\) is the smallest positive integer such that \(r(K_S+\Delta)\) is a Cartier divisor. Let \(\varphi\) be from the function field of \(S\) such that \(\text{div} (\varphi)=r(K_S+\Delta)\). The normalization \(\widetilde{S}\to S\) in the field extension defined by \(\sqrt[r]{\varphi}\) is called an index 1 cover associated to \(\varphi\). Index 1 covers \(\widetilde{S}\) of the purely \(\log\) terminal pair \((S, \Delta)\) are considered. Under the condition that \(S\) is smooth, char \((k)\geq 3\) and some other conditions it is proved that \(\widetilde{S}\) is canonical. In characteristic 2 this result is wrong. The counterexamples are given.