In algebraic geometry, the Kodaira dimensionκ(X) measures the size of the canonical model of a projective variety X.
Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation κ.[1] Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension),[2] and later named it after Kunihiko Kodaira.[3]
In algebraic geometry, the Kodairadimension κ(X) measures the size of the canonical model of a projective variety X. Igor Shafarevich in a seminar introduced...
surfaces, and families thereof, sorted according to their Kodairadimension following Enriques–Kodaira classification. Projective plane Cone (geometry) Cylinder...
redirection), follows: Examples of algebraic surfaces include (κ is the Kodairadimension): κ = −∞: the projective plane, quadrics in P3, cubic surfaces, Veronese...
the Kodairadimension of V. One can define an analogous ring for any line bundle L over V; the analogous dimension is called the Iitaka dimension. A line...
the Kodairadimension, which measures the growth of the plurigenera Pd as d goes to infinity. The Kodairadimension divides all varieties of dimension n...
In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodairadimension zero. A simple example...
rational normal curve. The Iitaka dimension of the canonical bundle of a smooth variety is called its Kodairadimension. Consider on complex algebraic varieties...
their Kodairadimension, as in the complex case. The four classes are: a) Kodairadimension minus infinity. These are the ruled surfaces. b) Kodaira dimension...
is an elliptic surface (with no singular fibers). All surfaces of Kodairadimension 1 are elliptic surfaces. Every complex Enriques surface is elliptic...
VII are non-algebraic complex surfaces studied by (Kodaira 1964, 1968) that have Kodairadimension −∞ and first Betti number 1. Minimal surfaces of class...
algebraic surface with Kodairadimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodairadimension 2 will actually...
theorem. The statement of Kunihiko Kodaira's result is that if M is a compact Kähler manifold of complex dimension n, L any holomorphic line bundle on...
mathematics, a Kodaira surface is a compact complex surface of Kodairadimension 0 and odd first Betti number. The concept is named after Kunihiko Kodaira. These...
polynomial equations than others. This is formalized by the notion of Kodairadimension of a variety, and by this measure projective spaces are the most special...
characteristic zero has Kodairadimension −∞. The converse is a conjecture which is known in dimension at most 3: a variety of Kodairadimension −∞ over a field...
non-singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodairadimension 0. In the early nineteenth century, the theory of...
simplicity is assumed non-singular. There are two cases based on its Kodairadimension, κ ( X ) {\displaystyle \kappa (X)} : κ ( X ) = − ∞ . {\displaystyle...
they are not unirational. They accomplished this by studying the Kodairadimension of the coarse moduli spaces κ g = K o d ( M ¯ g ) , {\displaystyle...
University working in algebraic geometry who introduced the Kodairadimension, and Iitaka dimension. He was a world leader in the field of Algebraic geometry...
fibrations induced by pluri-canonical systems on varieties of non-negative Kodairadimension. The problem consists of two halves: one related to general fibres...
In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds...
canonical ring or model can then be used to define the Kodairadimension of X. Projective schemes of dimension one are called projective curves. Much of the theory...
variety is finitely generated. The Kodairadimension of V is the dimension of the canonical ring minus one. Here the dimension of the canonical ring may be...
singularities Rational variety Unirational variety Ruled variety Kodairadimension Canonical ring Minimal model program Intersection theory Intersection...
one of the classes of surfaces of Kodairadimension 0 in the Enriques–Kodaira classification. The Kodairadimension is 0. Hodge diamond: Any hyperelliptic...
simply connected. A much easier fact is that every Fano variety has Kodairadimension −∞. Campana and Kollár–Miyaoka–Mori showed that a smooth Fano variety...
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