Double sections, dominating maps, and the Jacobian fibration

We give two parametrized versions of the uniformization theorem of a noncompact, non-hyperbolic Riemann surface using different but complementary methods. The first constructs the uniformizing maps directly in terms of coordinates via classical complex analysis and provides a canonical form for the double sections of a conic bundle over a noncompact complex curve. The second version, which is coordinate independent, works over any complex curve and is obtained by extending Kodaira's theory of the Jacobian fibration to a family of singular algebraic curves constructed via algebraic geometry. Then, using the results obtained with the Jacobian fibration, we give two equivalent conditions for a complex analytic surface nonhyperbolically fibered over a complex curve to be holomorphically dominable by [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]: We show that this dominability is equivalent to the apparently weaker condition of the existence of a Zariski dense image of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] and equivalent to the quasiprojectivity of the base curve together with the nonnegativity of the orbifold Euler characteristic. We discuss also the sharpness of our result in various contexts as well as the lack of connection to the fundamental group.