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 Maxwell Rosenlicht (1954), 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.

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.

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.

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