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Concept
Algebraic geometry of projective spaces
Summary
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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