LAUSR

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $$\mathcal {E}_{A,B}:\, A(x)-B(y)=0$$ , where $$A, B\in \mathbb {C}(z)$$ . We also investigate “series” of curves $$\mathcal {E}_{A,B}$$ of genus zero, where by a series we mean a family with the “same” A. We show that for a given rational function A a sequence of rational functions $$B_i$$ , such that $$\deg B_i\rightarrow \infty $$ and all the curves $$A(x)-B_i(y)=0$$ are irreducible and have genus zero, exists if and only if the Galois closure of the field extension $$\mathbb {C}(z)/\mathbb {C}(A)$$ has genus zero or one.