Wild Quotients of Products of Curves

Let k be an algebraically closed field of characteristic p > 0. Let B1/k and B2/k be two smooth proper connected curves, each endowed with an automorphism σi : Bi → Bi of order p. Let Y := B1 × B2, and let σ : Y → Y be the automorphism σ1 × σ2. We show that the graph of the resolution of the singularities of Y/ 〈σ〉 when B2 is an ordinary curve of positive genus is a star-shaped graph with three terminal chains. The intersection matrix N of the resolution satisfies | det(N)| = p, and can be completely determined when B1 is also ordinary, or when σ1 has a unique fixed point. Wild Z/pZ-quotient singularities of surfaces are expected to have resolution graphs which are trees, with associated intersection matricesN satisfying | det(N)| = p for some s ≥ 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 | det(N)| = p.