Fiber product of Riemann surfaces

If S 0 , 1 2 S_{0}, S_{1}, S_{2} are connected Riemann surfaces, alttext="beta colon right-arrow 0"> β : → encoding="application/x-tex">\beta _{1}:S_{1} \to S_{0} and 2 _{2}:S_{2} surjective holomorphic maps, then the associated fiber product times Subscript left-parenthesis beta right-parenthesis Baseline ×<!-- × stretchy="false">( stretchy="false">) encoding="application/x-tex">S_{1} \times _{(\beta _{1},\beta _{2})} has structure of a one-dimensional complex analytic space, endowed with canonical map : S_{1} S_{2} , such that, for alttext="j equals j = encoding="application/x-tex">j=1,2 j ring pi beta"> ∘<!-- ∘ <mml:mi>π<!-- π _{j} \circ \pi _{j}=\beta</mml:annotation> where alttext="pi Baseline"> encoding="application/x-tex">\pi _{j}: S_{j} is natural coordinate projection. The components complement its singular locus provide irreducible components. A Fuchsian description provided and, as consequence, we obtain that if one maps j"> _{j} regular branched covering, all isomorphic. Also, both 1"> _{1} _{2} finite degree, observe number these bounded above by greatest common divisor two degrees, an bound sharp. We also sufficient conditions irreducibility product. In case ModifyingAbove double-struck C With caret"> C ^<!-- ^ </mml:mover> encoding="application/x-tex">S_{0}=\widehat {\mathbb C} encoding="application/x-tex">S_{1} encoding="application/x-tex">S_{2} compact, define strong field moduli pair alttext="left-parenthesis right-parenthesis"> encoding="application/x-tex">(S_{1} S_{2},\beta ) this coincides minimal containing fields pairs encoding="application/x-tex">(S_{1},\beta _{1}) encoding="application/x-tex">(S_{2},\beta _{2}) . Finally, in surfaces encoding="application/x-tex">S_{0} compact surface, Jacobian variety J J encoding="application/x-tex">J(S_{1} _{2})}S_{2}) JS_{0} isogenous to P"> P encoding="application/x-tex">JS_{1} JS_{2} P encoding="application/x-tex">P suitable abelian subvariety encoding="application/x-tex">J (S_{1} _{2})}S_{2})