Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Double Sections and Dominating Maps   As is well-known, there exist nonconstant holomorphic maps from the plane
into the Riemann sphere $\PP^1$ 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
$\infty$ 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 $\infty$ 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 $\CC^2$. In theorem
\ref{thm:surf} we show that this dominability is equivalent to the apparently
weaker condition of the existence of a Zariski dense image of $\CC$ and
equivalent to the quasiprojectivity of the base curve together with the
nonnegativity of the orbifold Euler characteristic.

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The categories of this paper are: funct-an
19
The categories of this paper are: eess
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The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: stat