0 Double sections , dominating maps and the Jacobian fibration

As is well-known, there exist nonconstant holomorphic maps from the plane into the Riemann sphere P minus two points, the simplest example of which is an explicit realization of the uniformization map given by applying the exponential map and then composing with a Mobius transformation taking 0 and∞ to the two given punctures. Likewise, we can map the plane into the sphere minus one point by simply applying directly a Mobius transformation taking ∞ to this puncture. In this paper we prove two parametrized versions of this uniformization result using different methods. We first present a slightly easier version using complex analysis, which allows us to construct the uniformizing maps more or less explicitly in terms of the initial coordinates given. Then we give the more general and coordinate invariant version, which we prove by extending Kodaira’s theory of the Jacobian fibration to a family of singular curves constructed via algebraic geometry. Finally, using the results obtained with the Jacobian fibration, we obtain two equivalent conditions for an complex surface nonhyperbolically fibered over a complex curve to be holomorphically dominable by C. In theorem 1.7 we show that this dominability is equivalent to the apparently weaker condition of the existence of a Zariski dense image of C and equivalent to the quasiprojectivity of the base curve together with the nonnegativity of the orbifold Euler characteristic. We start with the following definition.