Prym variety

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In mathematics, the Prym variety construction is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it was applied to an unramified double covering of a Riemann surface, and was introduced by W. Schottky and H. W. E. Jung in relation with the Schottky problem, as it now called, of characterising Jacobian varieties among abelian varieties.

Given a non-constant morphism

φ: C₁ → C₂

of algebraic curves, write J_i for the Jacobian variety of C_i. Then from φ construct the corresponding morphism

ψ: J₁ → J₂,

which can be defined by taking the 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 of the reduced scheme underlying the scheme-theoretic kernel may be intended. Or in other words take the largest abelian subvariety of J₁, 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.