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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 used by F. Schottky and H. W. E. Jung in relation with the Schottky problem, 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, and was extensively studied by Wirtinger in 1895, including degenerate cases.
Given a non-constant morphism
of algebraic curves, write Ji for the Jacobian variety of Ci. Then from φ construct the corresponding morphism
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 of the reduced scheme underlying the kernel may be intended. Or in other words take the largest abelian subvariety of J1 on which ψ is trivial.