Classification up to equivalence of the holomorphic mappings of Riemann surfaces of low genus (Q542193)

Denote by \(\text{Hol}(S_{g}, S_{g'})\) the set of all holomorphic mappings of a Riemann surface \(S_{g}\) of genus \(g\) onto a Riemann surface \(S_{g'}\) of genus \(g'\), where \(g \geq g' > 1.\) De Franchis established that the cardinality of \(\text{Hol}(S_{g}, S_{g'})\) is finite and bounded above by a constant depending only on \(g.\) Two mappings \(f : S_{g} \rightarrow S_{g'}\) and \(h : S_{g} \rightarrow S_{g'}\) are called equivalent whenever there exist automorphisms \(\alpha \in \text{Aut}(S_{g})\) and \(\beta \in \text{Aut} (S_{g'})\) such that \(f \alpha = \beta h\). The goal of this article is to obtain a complete classification up to equivalence of the holomorphic mappings of a Riemann surface of genus 3 onto a Riemann surface of genus 2, in terms of algebraic curves and rational mappings. He also establish that every Riemann surface of genus 3 has at most three holomorphic images of genus 2.