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Double sections, dominating maps, and the Jacobian fibration. (English) Zbl 0967.32015
The authors give two parametrized versions of the uniformization theorem of a nonconstant, nonhyperbolic Riemann surface. The first constructs the uniformization map directly in terms of coordinates via classical complex analysis; the second one, 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.

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32S25 Complex surface and hypersurface singularities
14H55 Riemann surfaces; Weierstrass points; gap sequences
14J17 Singularities of surfaces or higher-dimensional varieties
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