Generalized Jacobian

In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by , and can be used to study ramified coverings of a curve, with abelian Galois group. Generalized Jacobians of a curve are extensions of the Jacobian of the curve by a commutative affine algebraic group, giving nontrivial examples of Chevalley's structure theorem.
Definition.
Suppose "C" is a complete nonsingular curve, "m" an effective divisor on "C", "S" is the support of "m", and "P" is a fixed base point on "C" not in "S". The generalized Jacobian "J""m" is a commutative algebraic group with a rational map "f" from "C" to "J""m" such that:
Moreover "J""m" is the universal group with these properties, in the sense that any rational map from "C" to a group with the properties above factors uniquely through "J""m". The group "J""m" does not depend on the choice of base point "P", though changing "P" changes that map "f" by a translation.
Structure of the generalized Jacobian.
For "m"=0 the generalized Jacobian "J""m" is just the usual Jacobian "J", an abelian variety of dimension "g", the genus of "C".
For "m" a nonzero effective divisor the generalized Jacobian is an extension of "J" by a connected commutative affine algebraic group "L""m" of dimension deg("m")−1. So we have an exact sequence
The group "L""m" is a quotient
of a product of groups "R""i" by the multiplicative group "G""m" of the underlying field. The product runs over the points "P""i" in the support of "m", and the group "R""i" is the group of invertible elements of the local ring modulo those that are 1 mod "m". The group "R""i" has dimension "n""i", the number of times "P""i" occurs in "m". It is the product of the multiplicative group "G""m" by a unipotent group of dimension "n""i"−1, which in characteristic 0 is isomorphic to a product of "n""i"−1 additive groups.
Complex generalized Jacobians.
Over the complex numbers, the algebraic structure of the generalized Jacobian determines an analytic structure of the generalized Jacobian making it a complex Lie group.
The analytic subgroup underlying the generalized Jacobian can be described as follows. (This does not always determine the algebraic structure as two non-isomorphic commutative algebraic groups may be isomorphic as analytic groups.) Suppose that "C" is a curve with an effective divisor "m" with support "S". There is a natural map from the homology group "H"1("C" − "S") to the dual Ω(−"m")* of the complex vector space Ω(−"m") (1-forms with poles on "m") induced by the integral of a 1-form over a 1-cycle. The analytic generalized Jacobian is then the quotient group Ω(−"m")*/"H"1("C" − "S").