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What can be said about a complex curve $C$, if its jacobian $J(C)$ has the maximal Picard number?

It is natural to expect that for a general curve of given genus its Jacobian has Picard rank 1 (isn't it?). What are the basic references for this fact?

Is it true, that there exists only finite number of curves of given genus, wich Jacobians are of the maximal Picard rank? If it is, then what is the reference?

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    $\begingroup$ A reference for the fact that the Jacobian of a very general curve has Picard number one should be found somewhere in: Birkenhake & Lange - Complex abelian varieties. A direct reference is also: Koizumi: The ring of algebraic correspondences on a generic curve of genus g. For your other question, there are some facts about curves with Jacobians having maximal Picartd number in: Beauville - Some surfaces with maximal Picard number. $\endgroup$
    – Bernie
    Commented Jul 18, 2017 at 9:26

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