To an abelian variety A over a field k, one associates a dual abelian variety A^v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a k-variety T is defined to be a line bundle L on A×T such that

1. for all t ∈ T, the restriction of L to A×{t} is a degree 0 line bundle,
2. the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).

Then there is a variety A^v and a line bundle P → A × A^v, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by A^v in the sense of the above definition. Moreover, this family is universal, that is, to any family L parametrized by T is associated a unique morphism f: T → A^v so that L is isomorphic to the pullback of P along the morphism 1_A×f: A × T → A × A^v. Applying this to the case when T is a point, we see that the points of A^v correspond to line bundles of degree 0 on A, so there is a natural group operation on A^v given by tensor product of line bundles, which makes it into an abelian variety.

Dual abelian variety is named after Niels Abel.