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- Prym_variety abstract "In mathematics, the Prym variety construction (named for Friedrich Prym) 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 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 morphismφ: C1 → C2of algebraic curves, write Ji for the Jacobian variety of Ci. Then from φ construct the corresponding morphismψ: J1 → J2,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.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. 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.".
- Prym_variety wikiPageID "4116892".
- Prym_variety wikiPageRevisionID "503945963".
- Prym_variety hasPhotoCollection Prym_variety.
- Prym_variety subject Category:Abelian_varieties.
- Prym_variety subject Category:Algebraic_curves.
- Prym_variety type AbelianVarieties.
- Prym_variety type Abstraction100002137.
- Prym_variety type AlgebraicCurves.
- Prym_variety type Assortment108398773.
- Prym_variety type Attribute100024264.
- Prym_variety type Collection107951464.
- Prym_variety type Curve113867641.
- Prym_variety type Group100031264.
- Prym_variety type Line113863771.
- Prym_variety type Shape100027807.
- Prym_variety comment "In mathematics, the Prym variety construction (named for Friedrich Prym) 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 now called, of characterising Jacobian varieties among abelian varieties.".
- Prym_variety label "Prym variety".
- Prym_variety sameAs m.0bk027.
- Prym_variety sameAs Q7253243.
- Prym_variety sameAs Q7253243.
- Prym_variety sameAs Prym_variety.
- Prym_variety wasDerivedFrom Prym_variety?oldid=503945963.
- Prym_variety isPrimaryTopicOf Prym_variety.