In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
a generalized Jacobian is a commutative
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
associated to a curve with a divisor, generalizing the
Jacobian variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...
of a complete curve. They were introduced by
Maxwell Rosenlicht
Maxwell Alexander Rosenlicht (April 15, 1924 – January 22, 1999) was an American mathematician known for works in algebraic geometry, algebraic groups, and differential algebra.
Rosenlicht went to school in Brooklyn ( Erasmus High School) and ...
in 1954, and can be used to study
ramified covering
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
s of a curve, with abelian
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
. Generalized Jacobians of a curve are extensions of the Jacobian of the curve by a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
affine algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...
, giving nontrivial examples of
Chevalley's structure theorem
In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety
In mathematics, ...
.
Definition
Suppose ''C'' is a complete
nonsingular
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
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
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
of
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
''g'', the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
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
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
.
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
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
''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