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Picard group information


In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.

Alternatively, the Picard group can be defined as the sheaf cohomology group

For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group.

The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces.

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Picard group

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Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group...

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Picard

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Look up Picard or picard in Wiktionary, the free dictionary. Picard may refer to: Picardy, a region of France Picard language, a language of France Jean-Luc...

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Invertible sheaf

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Vector bundles in algebraic geometry Line bundle First Chern class Picard group Birkhoff-Grothendieck theorem EGA 0I, 5.4. Stacks Project, tag 01CR,...

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Picard modular group

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In mathematics, a Picard modular group, studied by Picard (1881), is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of integers...

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Ring theory

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abelian group called the Picard group of R. If R is an integral domain with the field of fractions F of R, then there is an exact sequence of groups: 1 →...

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K3 surface

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theory of curves on a surface, by identifying the Picard group with the divisor class group.) The Picard lattice of a K3 surface is always even, meaning...

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Holomorphic vector bundle

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geometry, the Picard group Pic(X) of the complex manifold X is the group of isomorphism classes of holomorphic line bundles with group law given by tensor...

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Adele ring

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profinite completion of the group. If X / F q {\displaystyle X/\mathbf {F_{\mathit {q}}} } is a smooth proper curve then its Picard group is Pic ( X )   =   K...

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Jacobian variety

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0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety. The Jacobian variety is named after Carl...

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Picard language

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Picard (/ˈpɪkɑːrd/, also US: /pɪˈkɑːrd, ˈpɪkərd/, French: [pikaʁ] ) is a langue d'oïl of the Romance language family spoken in the northernmost of France...

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Berger Picard

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The Berger Picard (/bɛərˌʒeɪ pɪˈkɑːr/ bair-ZHAY pih-KAR, French: [bɛʁʒe pikaʁ]) or Picardy Shepherd, is a French herding dog originating in Picardy. These...

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Rosalind Picard

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Rosalind Wright Picard (born May 17, 1962) is an American scholar and inventor who is Professor of Media Arts and Sciences at MIT, founder and director...

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Tautological bundle

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bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space. In Michael Atiyah's "K-theory", the tautological...

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Del Pezzo surface

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Picard group of a del Pezzo surface of degree d is the odd unimodular lattice I1,9−d, except when the surface is a product of 2 lines when the Picard...

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Fuchsian group

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of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called Picard group PSL(2,Z[i]) is...

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Exponential sheaf sequence

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i\,\mathbb {Z} )\to \cdots .} Here H1(OM*) can be identified with the Picard group of holomorphic line bundles on M. The connecting homomorphism sends a...

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Ideal class group

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Theorem Narrow class group Picard group—a generalisation of the class group appearing in algebraic geometry Arakelov class group Claborn 1966 Neukirch...

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Ernest Picard

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government standpoint, Picard, one of the members of the group known as Les Cinq, veered more to the left. In the 1860s Picard was an active member of...

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