Algebraic geometry of projective spaces

The 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. The projective Nullstellensatz states that, for any homogeneous ideal I that does not contain all polynomials of a certain degree (referred to as an irrelevant ideal), the common zero locus of all polynomials in I (or Nullstelle) is non-trivial (i.e. the common zero locus contains more than the single element {0}), and, more precisely, the ideal of polynomials that vanish on that locus coincides with the radical of the ideal I. This last assertion is best summarized by the formula : for any relevant ideal I, In particular, maximal homogeneous relevant ideals of k[V] are one-to-one with lines through the origin of V. Let V be a finite-dimensional vector space over a field k. The scheme over k defined by Proj(k[V]) is called projectivization of V. The projective n-space on k is the projectivization of the vector space . The definition of the sheaf is done on the base of open sets of principal open sets D(P), where P varies over the set of homogeneous polynomials, by setting the sections to be the ring , the zero degree component of the ring obtained by localization at P. Its elements are therefore the rational functions with homogeneous numerator and some power of P as the denominator, with same degree as the numerator. The situation is most clear at a non-vanishing linear form φ. The restriction of the structure sheaf to the open set D(φ) is then canonically identified with the affine scheme spec(k[ker φ]). Since the D(φ) form an open cover of X the projective schemes can be thought of as being obtained by the gluing via projectivization of isomorphic affine schemes.

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Euler sequence

In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an -fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Let be the n-dimensional projective space over a commutative ring A. Let be the sheaf of 1-differentials on this space, and so on.

Glossary of algebraic geometry

This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.

Tautological bundle

In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles.

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