The relative Riemann–Hurwitz formula. Issue 12 (2nd December 2022) - Ding, Zhiguo; Zieve, Michael E.
Abstract: For any nonconstant f, g ∈ C ( x ) such that the numerator H ( x, y ) of f ( x ) − g ( y ) is irreducible, we compute the genus of the normalization of the curve H ( x, y ) = 0 . We also prove an analogous formula in arbitrary characteristic when f and g have no common wildly ramified branch points, and generalize to (possibly reducible) fiber products of nonconstant morphisms of curves f : A → D and g : B → D .