Generalized Jacobian

Last updated November 09, 2023

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 in 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 → ΠU_{P_i}^(n_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 U_{P_i}^(n_i) is the group of invertible elements of the local ring modulo those that are 1 mod P_i^n_i. The group U_{P_i}^(n_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 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).

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References

Rosenlicht, Maxwell (1954), "Generalized Jacobian varieties.", Ann. of Math., 2, 59 (3): 505–530, doi:10.2307/1969715, JSTOR 1969715, MR 0061422
Serre, Jean-Pierre (1988) [1959], Algebraic groups and class fields. , Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag, ISBN 0-387-96648-X, MR 0103191

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