What is the degree of a vector bundle?
We want to give a definition of degree for vector bundles. With the correspondence between divisors and line bundles, we define the degree of a line bundle as the degree of its associated divisor.
What is a positive line bundle?
A holomorphic line bundle on a complex manifold is called positive if its curvature differential 2-form is, after multiplication with i=√−1, a positive definite (1,1)-form.
What is an ample line bundle?
A line bundle is ample if some positive power is very ample. An ample line bundle on a projective variety X has positive degree on every curve in X. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness.
What is a trivial bundle?
A bundle or fiber bundle is trivial if it is isomorphic to the cross product of the base space and a fiber.
What is a section of a vector bundle?
A bundle section of a vector bundle is a map whose projection, is the identity map on . For instance, on a trivial bundle , a section corresponds to a function by . Near every point in a vector bundle, there is a trivialization. The structure of the vector bundle, as in all bundles, is that it is locally trivial.
What does ample mean?
Definition of ample 1 : generous or more than adequate in size, scope, or capacity There was room for an ample garden. 2 : generously sufficient to satisfy a requirement or need They had ample money for the trip.
Is a vector bundle a fiber bundle?
A special class of fiber bundles, called vector bundles, are those whose fibers are vector spaces (to qualify as a vector bundle the structure group of the bundle — see below — must be a linear group). Important examples of vector bundles include the tangent bundle and cotangent bundle of a smooth manifold.
What is a fiber bundle math?
A fiber bundle (also called simply a bundle) with fiber is a map where is called the total space of the fiber bundle and the base space of the fiber bundle. The main condition for the map to be a fiber bundle is that every point in the base space has a neighborhood such that is homeomorphic to in a special way.
What is a bundle in math?
Bundling is also called grouping. This is a way to group numbers by putting the smaller units together to make a larger one. For instance, putting 10 ones together makes 1 ten. Putting 10 tens together makes 1 hundred. Counting using Base Ten.
What is a Canuckle?
The brainchild of Ottawa resident, Mark Rogers, Canuckle is a Wordle-based guessing game that challenges the player to guess a five-letter word that in some way relates to Canada. The mystery word could be anything from a word, place, or simple Canadianism – if it’s tied to Canada, it goes.
What are Bussums?
1a : the human chest and especially the front part of the chest hugged the child to his bosom. b : a woman’s breasts regarded especially as a single feature a woman with an ample bosom also : breast a woman’s bosoms.
Are bundles manifolds?
Differentiable fiber bundles is called a fibered manifold.
What is the degree of a line bundle on a curve?
The degree of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole. Therefore, any line bundle L on a curve C such that
What is a line bundle in math?
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these.
When is a line bundle ample?
More strongly, a line bundle on X is very ample if it has enough sections to give a closed immersion (or “embedding”) of X into projective space. A line bundle is ample if some positive power is very ample.
When is a line bundle globally generated?
A line bundle is globally generated if and only if it is basepoint-free. For example, every quasi-coherent sheaf on an affine scheme is globally generated. Analogously, in complex geometry, Cartan’s theorem A says that every coherent sheaf on a Stein manifold is globally generated.