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This chapter contains the construction and elementary study of the generalized Jacobians of an algebraic curve. We will follow closely the paper of Rosenlicht  on this subject, itself inspired by Weil’s Variétés abéliennes , where the case of the usual Jacobian is treated. We will make use, as they did, of the method of “generic points”. This obliges us to renounce the point of view of the preceding chapters (where all points had their coordinates in a fixed base field), and to adopt that of Foundations . It is certain that the generic points could be replaced by divisors on product varieties, after first developing in detail the properties of these divisors (that is to say essentially the cohomology of coherent algebraic sheaves on a product variety); that would take us too far afield.
KeywordsAlgebraic Group Galois Group Base Field Symmetric Product Effective Divisor
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