Mathematical group occurring in algebraic geometry and the theory of complex manifolds
Not to be confused with Picard modular group.
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.
Picardgroup 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...
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...
abelian group called the Picardgroup 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 →...
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...
geometry, the Picardgroup Pic(X) of the complex manifold X is the group of isomorphism classes of holomorphic line bundles with group law given by tensor...
theory of curves on a surface, by identifying the Picardgroup with the divisor class group.) The Picard lattice of a K3 surface is always even, meaning...
The Berger Picard (/bɛərˌʒeɪ pɪˈkɑːr/ bair-ZHAY pih-KAR, French: [bɛʁʒe pikaʁ]) or Picardy Shepherd, is a French herding dog originating in Picardy. These...
profinite completion of the group. If X / F q {\displaystyle X/\mathbf {F_{\mathit {q}}} } is a smooth proper curve then its Picardgroup is Pic ( X ) = K...
0 line bundles. It is the connected component of the identity in the Picardgroup of C, hence an abelian variety. The Jacobian variety is named after Carl...
Picard (/ˈpɪkɑːrd/, also US: /pɪˈkɑːrd, ˈpɪkərd/, French: [pikaʁ] ) is a langue d'oïl of the Romance language family spoken in the northernmost of France...
A Picard horn, also called the Picard topology or Picard model, is one of the oldest known hyperbolic 3-manifolds, first described by Émile Picard in 1884...
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...
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...
bundle and the hyperplane bundle are exactly the two generators of the Picardgroup of the projective space. In Michael Atiyah's "K-theory", the tautological...
Picardgroup 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...
of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called Picardgroup PSL(2,Z[i]) is...
i\,\mathbb {Z} )\to \cdots .} Here H1(OM*) can be identified with the Picardgroup of holomorphic line bundles on M. The connecting homomorphism sends a...
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