mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 Humboldt University of Berlin, University of Berli ...

) is a method in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

of making an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

from a

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s. 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 c ...

, as it is now called, of characterising Jacobian varieties among abelian varieties. It is said to have appeared first in the late work of

Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...

, 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 Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

of ψ. To qualify that somewhat, to get an abelian ''variety'', the connected component of the identity of the

reduced scheme This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geomet ...

underlying the

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 David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded th ...

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 Kadomtsev, Boris Borisovich Kadomtsev and Vladimir Iosifovi ...

. 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 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