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 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'' → Π''U''_''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 ''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'').

References

*
*{{citation, mr= 0103191, last= Serre, first= Jean-Pierre, title= Algebraic groups and class fields., series= Graduate Texts in Mathematics, volume= 117, publisher= Springer-Verlag, place= New York, year= 1988, isbn= 0-387-96648-X, orig-year= 1959, url-access= registration, url= https://archive.org/details/algebraicgroupsc0000serr

Algebraic groups
Algebraic curves