Ample line bundleIn mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space.
Coherent sheafIn mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an , and so they are closed under operations such as taking , , and cokernels.
Characteristic classIn mathematics, a characteristic class is a way of associating to each principal bundle of X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry.
Algebraic geometry of projective spacesThe concept of a Projective space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective spaces. Let k be an algebraically closed field, and V be a finite-dimensional vector space over k. The symmetric algebra of the dual vector space V* is called the polynomial ring on V and denoted by k[V]. It is a naturally graded algebra by the degree of polynomials.
Divisor (algebraic geometry)In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1.
Line bundleIn 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. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner.
Real projective spaceIn mathematics, real projective space, denoted \mathbb{RP}^n or \mathbb{P}_n(\R), is the topological space of lines passing through the origin 0 in the real space \R^{n+1}. It is a compact, smooth manifold of dimension n, and is a special case \mathbf{Gr}(1, \R^{n+1}) of a Grassmannian space. As with all projective spaces, RPn is formed by taking the quotient of Rn+1 ∖ under the equivalence relation x ∼ λx for all real numbers λ ≠ 0. For all x in Rn+1 ∖ one can always find a λ such that λx has norm 1.
Invertible sheafIn mathematics, an invertible sheaf is a sheaf on a ringed space which has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Let (X, OX) be a ringed space. Isomorphism classes of sheaves of OX-modules form a monoid under the operation of tensor product of OX-modules. The identity element for this operation is OX itself.
