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

Let S be a normal surface over an algebraically closed field k and let be a standard boundary. We consider index 1 covers of the purely log terminal pair (S,). We prove that when S is smooth and char k=p3, then is canonical under some conditions. To prove this, we classify the boundary =(1–1/b_i)D_i which makes (S,) a purely log terminal pair, and then reduce equations defining singularity of to the normal forms of RDP. Unfortunately there are some counterexamples in p=2, and we classify them. These results give partial solutions to the index 1 cover conjecture in positive characteristic.Mathematical Subject Classification (2000): 14E20