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.

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:

• f takes P to the identity of J_m.
• f is regular outside S.
• f(D) = 0 whenever D is the divisor of a rational function g on C such that g≡1 mod m.

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

0 → L_m → J_m → J → 0

The group L_m is a quotient

0 → G_m → ΠR_i → L_m → 0

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 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 H₁(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)*/H₁(C−S).

References

• Rosenlicht, Maxwell (1954), "Generalized Jacobian varieties.", Ann. of Math. (2), 59: 505–530, doi:10.2307/1969715, JSTOR 1969715, MR 0061422 
• Serre, Jean-Pierre (1988) [1959], Algebraic groups and class fields., Graduate Texts in Mathematics, 117, New York: Springer-Verlag, ISBN 0-387-96648-X, MR 0103191