On the fiber product of Riemann surfaces

Let $S_0,S_1$ and $S_2$ be connected Riemann surfaces and let ${\beta }_1:S_1\to S_0$ and ${\beta }_2:S_2\to S_0$ be surjective holomorphic maps. The associated fiber product $S_1{\times }_{({\beta }_1,{\beta }_2)}{S}_2$ has the structure of a singular Riemann surface, endowed with a canonical map $\beta$ to $S_0$ satisfying that ${\beta }_j\circ {\pi }_j=\beta$, where ${\pi }_j$ is coordinate projection onto $S_j$. In this paper we provide a Fuchsian description of the fiber product and obtain that if one the maps ${\beta }_j$ is a regular branched cover, then all its irreducible components are isomorphic. In the case that both ${\beta }_j$ are of finite degree, we observe that the number of irreducible components is bounded above by the greatest common divisor of the two degrees; we study the irreducibility of the fiber product. In the case that $S_0=\widehat{\mathbb{C}}$, and $S_1$ and $S_2$ are compact, we define the strong field of moduli of the pair $(S_{1} \times_{(\beta_{1},\beta_{2})} S_{2},\beta)$ and observe that this field coincides with the minimal field containing the fields of moduli of both pairs $(S_1,{\beta }_1)$ and $(S_2,{\beta }_2)$. Finally, in the case that the fiber product is a connected Riemann surface, we provide an isogenous decomposition of its Jacobian variety.