Wild quotients of products of curves

Abstract

Let k be an algebraically closed field of characteristic $$p>0$$ . Let and be two smooth proper connected curves, each endowed with an automorphism $$\sigma _i:B_i \rightarrow B_i$$ of order p. Let , and let $$\sigma :Y \rightarrow Y$$ be the automorphism . We show that the graph of the resolution of any singularity of is a star-shaped graph with three terminal chains when $$B_2$$ is an ordinary curve of positive genus. The intersection matrix N of the resolution satisfies , and can be completely determined when $$B_1$$ is also ordinary, or when $$\sigma _1$$ has a unique fixed point. The singularity is rational. Wild -quotient singularities of surfaces are expected to have resolution graphs which are trees, with associated intersection matrices N satisfying for some $$r\geqslant 0$$ . We show, for any $$s>0$$ coprime to p, the existence of resolution graphs with one node, $$s+2$$ terminal chains, and with intersection matrix N satisfying .