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- Symmetric_product_of_an_algebraic_curve abstract "In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian productC × C × ... × Cor Cn by the group action of the symmetric group on n letters permuting the factors. It exists as a smooth algebraic variety ΣnC; if C is a compact Riemann surface it is therefore a complex manifold. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients.For C the projective line (say the Riemann sphere) ΣnC can be identified with projective space of dimension n.If G has genus g ≥ 1 then the ΣnC are closely related to the Jacobian variety J of C. More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor) have dimension n and fill up J, with some identifications caused by special divisors.For g = n we have ΣgC actually birationally equivalent to J; the Jacobian is a blowing down of the symmetric product. That means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil's technique of constructing J as an abstract variety from 'birational data'. Other ways of constructing J, for example as a Picard variety, are preferred now (Greg W. Anderson (Advances in Math.172 (2002) 169–205) provided an elementary construction as lines of matrices). But this does mean that for any rational function F on CF(x1) + ... + F(xg)makes sense as a rational function on J, for the xi staying away from the poles of F.For N > g the mapping from ΣnC to J by addition fibers it over J; when n is large enough (around twice g) this becomes a projective space bundle (the Picard bundle). It has been studied in detail, for example by Kempf and Mukai.".
- Symmetric_product_of_an_algebraic_curve wikiPageID "3140914".
- Symmetric_product_of_an_algebraic_curve wikiPageRevisionID "597641474".
- Symmetric_product_of_an_algebraic_curve hasPhotoCollection Symmetric_product_of_an_algebraic_curve.
- Symmetric_product_of_an_algebraic_curve subject Category:Algebraic_curves.
- Symmetric_product_of_an_algebraic_curve subject Category:Symmetric_functions.
- Symmetric_product_of_an_algebraic_curve type Abstraction100002137.
- Symmetric_product_of_an_algebraic_curve type AlgebraicCurves.
- Symmetric_product_of_an_algebraic_curve type Attribute100024264.
- Symmetric_product_of_an_algebraic_curve type Curve113867641.
- Symmetric_product_of_an_algebraic_curve type Function113783816.
- Symmetric_product_of_an_algebraic_curve type Line113863771.
- Symmetric_product_of_an_algebraic_curve type MathematicalRelation113783581.
- Symmetric_product_of_an_algebraic_curve type Relation100031921.
- Symmetric_product_of_an_algebraic_curve type Shape100027807.
- Symmetric_product_of_an_algebraic_curve type SymmetricFunctions.
- Symmetric_product_of_an_algebraic_curve comment "In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian productC × C × ... × Cor Cn by the group action of the symmetric group on n letters permuting the factors. It exists as a smooth algebraic variety ΣnC; if C is a compact Riemann surface it is therefore a complex manifold.".
- Symmetric_product_of_an_algebraic_curve label "Symmetric product of an algebraic curve".
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- Symmetric_product_of_an_algebraic_curve sameAs Q7661314.
- Symmetric_product_of_an_algebraic_curve sameAs Q7661314.
- Symmetric_product_of_an_algebraic_curve sameAs Symmetric_product_of_an_algebraic_curve.
- Symmetric_product_of_an_algebraic_curve wasDerivedFrom Symmetric_product_of_an_algebraic_curve?oldid=597641474.
- Symmetric_product_of_an_algebraic_curve isPrimaryTopicOf Symmetric_product_of_an_algebraic_curve.