In mathematics, the Jacobian variety of a non-singular algebraic curve C of genus g ≥ 1 is a particular abelian variety J, of dimension g. The curve C is a subvariety of J, and generates J as a group. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... :For other senses of this word, see dimension (disambiguation). ... In botanical nomenclature, a subvariety (subvarietas) is a taxon at a rank below that of variety (varietas) but above that of form (forma): it is an infraspecific taxon. ...

Analytically, it can be realized as the quotient space V/L, where V is the vector space of all In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

where γ is a path in C(C), and L is the lattice of all those l with closed path γ. In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ... In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ...

An important theorem regarding Jacobian varieties is Abel's theorem. Look up theorem in Wiktionary, the free dictionary. ... In mathematics, the Abel-Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. ...

References

• J.S. Milne (1986). "Jacobian Varieties". Arithmetic Geometry: pp. 167-212, New York: Springer-Verlag. ISBN 0-387-96311-1.