projective variety called the of V. If defines V, then the defining ideal of this closure is the homogeneous ideal of generated by for all f in I. For example, if V is an affine curve given by, say, in the affine plane, then its projective completion in the projective plane is given by Projective schemes For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., is a scheme which it is a union of (n + 1) copies of the affine n-space kn. More generally, projective space over a ring A is the union of the affine schemes in such a way the variables match up as expected. The set of closed points of , for algebraically closed fields k, is then the projective space in the usual sense. An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme. For example, if A is a ring, then If R is a quotient of by a homogeneous ideal I, then the canonical surjection induces the closed immersion Compared to projective varieties, the condition that the ideal I be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the topological space may have multiple irreducible components. Moreover, there may be nilpotent functions on X. Closed subschemes of correspond bijectively to the homogeneous ideals I of that are saturated; i.e., This fact may be considered as a refined version of projective Nullstellensatz. We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space V over k, we let where is the symmetric algebra of . It is the projectivization of V; i.e., it parametrizes lines in V. There is a canonical surjective map , which is defined using the chart described above. One important use of the construction is this (cf., ). A divisor D on a projective variety X corresponds to a line bundle L. One then set ; it is called the complete linear system of D. Projective space over any scheme S can be defined as a fiber product of schemes