Jacobian variety - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Mathematics, genus has a few different but closely related meanings Topology Orientable surface In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it The curve C is a subvariety of J, and generates J as a group. In botanical nomenclature a subvariety ( subvarietas) is a Taxonomic rank below that of variety ( varietas) but above that of form In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

Analytically, it can be realized as the quotient space V/L, where V is the vector space of all

$l = \int_{\gamma} (\cdot): \{\mbox{rational differentials on } C \mbox{ without poles}\} \longrightarrow \mathbb{C}, \quad \omega \mapsto \int_{\gamma} \omega$

where γ is a path in C(C), and L is the lattice of all those l with closed path γ. In Mathematics, algebraic geometry and analytic geometry are two closely related subjects In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to

An important theorem regarding Jacobian varieties is Abel's theorem. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements 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.

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