Riemann–Hurwitz theorem and Riemann–Roch theorem for hypermaps

Abstract

In this paper, we try to answer some questions raised by Cangelmi (Eur J Comb 33(7):1444–1448, 2012). We reinterpret the Riemann–Hurwitz theorem of orientable algebraic hypermaps by introducing tripartite graph morphisms and obtain Riemann–Roch theorems for orientable hypermaps by defining the divisor of a function f on darts. In addition, we extend Riemann–Roch theorem to non-orientable hypermaps by suitably replacing the orientable genus with the non-orientable genus. Finally, as an application of the Riemann–Hurwitz theorem, we establish the second main theorem from the viewpoint of Nevanlinna theory.