Canonical Ring

Canonical Ring

In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is the graded ring

of sections of powers of the canonical bundle K. Its nth graded component (for ) is:

that is, the space of sections of the n-th tensor product Kn of the canonical bundle K.

The 0th graded component is sections of the trivial bundle, and is one dimensional as V is projective. The projective variety defined by this graded ring is called the canonical model of V, and the dimension of the canonical model, is called the Kodaira dimension 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 bundle is called big if the Iitaka dimension equals the dimension of the variety.

Read more about Canonical Ring:  The Plurigenera

Famous quotes containing the words canonical and/or ring:

    If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.
    Thomas Hobbes (1579–1688)

    But whatever happens, wherever the scene is laid, somebody, somewhere, will quietly set out—somebody has already set out, somebody still rather far away is buying a ticket, is boarding a bus, a ship, a plane, has landed, is walking toward a million photographers, and presently he will ring at my door—a bigger, more respectable, more competent Gradus.
    Vladimir Nabokov (1899–1977)