The relative Riemann–Hurwitz formula. Issue 12 (2nd December 2022)

• Record Type: Journal Article
• Authors: Ding, Zhiguo; Zieve, Michael E.
• Is Part Of: Communications in algebra. Volume 50:Issue 12(2022)
• Page Start: 5033
• Page End: 5041
• Publication Date: 2022-12-02
• Subjects: Algebraic curve -- fiber product -- genus -- Riemann-Hurwitz -- separated variables; 14H45; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.1968886
• Languages: English
• ISSNs: 0092-7872

Abstract

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 .