The relative Riemann-Hurwitz formula

TL;DR

This paper derives a formula for the genus of certain algebraic curves defined by rational functions, extending classical results to arbitrary characteristic and more complex fiber products.

Contribution

It introduces a relative Riemann-Hurwitz formula applicable to a broad class of algebraic curves and their fiber products, generalizing existing genus computations.

Findings

Computed genus of normalization of H(x,y)=0 for irreducible H

Extended formula to arbitrary characteristic with no wild ramification

Generalized to fiber products of morphisms of curves

Abstract

For any nonconstant f,g in 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.