mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Prym variety construction (named for

Friedrich Prym Friedrich Emil Fritz Prym (28 September 1841, Düren – 15 December 1915, Bonn) was a German mathematician who introduced Prym varieties and Prym differentials. Prym completed his Ph.D. at the University of Berlin in 1863 with a thesis writt ...

) is a method 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 ...

of making 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 ...

from a

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

of algebraic curves. In its original form, it was applied to an unramified double covering of a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

, and was used by F. Schottky and H. W. E. Jung in relation with the

Schottky problem In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties. Geometric formulation More precisely, one should co ...

, as it is now called, of characterising

Jacobian varieties 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 ...

among abelian varieties. It is said to have appeared first in the late work of Riemann, and was extensively studied by Wirtinger in 1895, including degenerate cases. Given a non-constant morphism :φ: ''C''₁ → ''C''₂ of algebraic curves, write ''J''_''i'' for the Jacobian variety of ''C''_''i''. Then from φ construct the corresponding morphism :ψ: ''J''₁ → ''J''₂, which can be defined on a divisor class ''D'' of degree zero by applying φ to each point of the divisor. This is a well-defined morphism, often called the ''norm homomorphism''. Then the Prym variety of φ is the kernel of ψ. To qualify that somewhat, to get an abelian ''variety'', the

connected component of the identity In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compo ...

of the reduced scheme underlying the kernel may be intended. Or in other words take the largest abelian subvariety of ''J''₁ on which ψ is trivial. The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970. It now plays a substantial role in some contemporary theories, for example of the

Kadomtsev–Petviashvili equation In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili ...

. One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians. For example,

principally polarized 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 functio ...

abelian varieties (p.p.a.v.'s) of dimension > 3 are not generally Jacobians, but all p.p.a.v.'s of dimension 5 or less are Prym varieties. It is for this reason that p.p.a.v.'s are fairly well understood up to dimension 5.

References

* * {{Algebraic curves navbox Algebraic curves Abelian varieties