Algebraic geometry of projective spaces
Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space.
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 .
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