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Let k be an algebraically closed field of characteristic $$p>0$$ p>0 . Let and be two smooth proper connected curves, each endowed with an automorphism $$\sigma _i:B_i \rightarrow B_i$$ σi:Bi→Bi of order p. Let , and let $$\sigma :Y \rightarrow Y$$ σ:Y→Y be the automorphism . We show that the graph of the resolution of any singularity of is a star-shaped graph with three terminal chains...