Editing Projective variety (section)

== Variety and scheme structure ==

=== Variety structure ===
Let ''k'' be an algebraically closed field. The basis of the definition of projective varieties is projective space <math>\mathbb{P}^n</math>, which can be defined in different, but equivalent ways:

* as the set of all lines through the origin in <math>k^{n+1}</math> (i.e., all one-dimensional vector subspaces of <math>k^{n+1}</math>)
* as the set of tuples <math>(x_0, \dots, x_n) \in k^{n+1}</math>, with <math>x_0, \dots, x_n</math> not all zero, modulo the equivalence relation <math display="block">(x_0, \dots, x_n) \sim \lambda (x_0, \dots, x_n)</math> for any <math>\lambda \in k \setminus \{ 0 \}</math>. The equivalence class of such a tuple is denoted by <math display="block">[x_0: \dots: x_n].</math> This equivalence class is the general point of projective space. The numbers <math>x_0, \dots, x_n</math> are referred to as the [[homogeneous coordinates]] of the point.

A ''projective variety'' is, by definition, a closed subvariety of <math>\mathbb{P}^n</math>, where closed refers to the [[Zariski topology]].<ref>{{citation|author=Shafarevich|first=Igor R.|author-link=Igor Shafarevich|title=Basic Algebraic Geometry 1: Varieties in Projective Space|publisher=Springer|year=1994}}</ref> In general, closed subsets of the Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions. Given a polynomial <math>f \in k[x_0, \dots, x_n]</math>, the condition

:<math>f([x_0: \dots: x_n]) = 0</math>

does not make sense for arbitrary polynomials, but only if ''f'' is [[homogeneous polynomial|homogeneous]], i.e., the degrees of all the [[monomial]]s (whose sum is ''f'') are the same. In this case, the vanishing of

:<math>f(\lambda x_0, \dots, \lambda x_n) = \lambda^{\deg f} f(x_0, \dots, x_n)</math>

is independent of the choice of <math>\lambda \ne 0</math>.

Therefore, projective varieties arise from homogeneous [[prime ideal]]s ''I'' of <math>k[x_0, \dots, x_n]</math>, and setting

:<math>X = \left\{[x_0: \dots: x_n] \in \mathbb{P}^n, f([x_0: \dots: x_n]) = 0 \text{ for all }f \in I \right\}.</math>

Moreover, the projective variety ''X'' is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of ''X'' (e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective space <math>\mathbb{P}^n</math> is covered by the standard open affine charts

:<math>U_i = \{[x_0: \dots: x_n], x_i \ne 0 \},</math>

which themselves are affine ''n''-spaces with the coordinate ring

:<math>k \left [y^{(i)}_1, \dots, y^{(i)}_n \right ], \quad y^{(i)}_j = x_j/x_i.</math>

Say ''i'' = 0 for the notational simplicity and drop the superscript (0). Then <math>X \cap U_0</math> is a closed subvariety of <math>U_0 \simeq \mathbb{A}^n</math> defined by the ideal of <math>k[y_1, \dots, y_n]</math> generated by

:<math>f(1, y_1, \dots, y_n)</math>

for all ''f'' in ''I''. Thus, ''X'' is an algebraic variety covered by (''n''+1) open affine charts <math>X \cap U_i</math>.

Note that ''X'' is the closure of the affine variety <math>X \cap U_0</math> in <math>\mathbb{P}^n</math>. Conversely, starting from some closed (affine) variety <math>V \subset U_0 \simeq \mathbb{A}^n</math>, the closure of ''V'' in <math>\mathbb{P}^n</math> is the projective variety called the '''{{vanchor|projective completion}}''' of ''V''. If <math>I \subset k[y_1, \dots, y_n]</math> defines ''V'', then the defining ideal of this closure is the homogeneous ideal<ref>This homogeneous ideal is sometimes called the homogenization of ''I''.</ref> of <math>k[x_0, \dots, x_n]</math> generated by

:<math>x_0^{\deg(f)} f(x_1/x_0, \dots, x_n/x_0)</math>

for all ''f'' in ''I''.

For example, if ''V'' is an affine curve given by, say, <math>y^2 = x^3 + ax + b</math> in the affine plane, then its projective completion in the projective plane is given by <math>y^2 z = x^3 + ax z^2 + b z^3.</math>

=== 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., <math>\mathbb{P}^n(k)</math> is a scheme which it is a union of (''n'' + 1) copies of the affine ''n''-space ''k<sup>n</sup>''. More generally,<ref>{{harvnb|Mumford|1999|loc=pg. 82}}</ref> projective space over a ring ''A'' is the union of the [[affine scheme]]s

:<math>U_i = \operatorname{Spec} A[x_0/x_i, \dots, x_n/x_i], \quad 0 \le i \le n,</math>

in such a way the variables match up as expected. The set of [[closed point]]s of <math>\mathbb{P}^n_k</math>, for algebraically closed fields ''k'', is then the projective space <math>\mathbb{P}^n(k)</math> 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.<ref>{{harvnb|Hartshorne|1977|loc=Section II.5}}</ref> For example, if ''A'' is a ring, then

:<math>\mathbb{P}^n_A = \operatorname{Proj}A[x_0, \ldots, x_n].</math>

If ''R'' is a [[Quotient ring|quotient]] of <math>k[x_0, \ldots, x_n]</math> by a homogeneous ideal ''I'', then the canonical surjection induces the [[closed immersion]]

:<math>\operatorname{Proj} R \hookrightarrow \mathbb{P}^n_k.</math>

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]] <math>X = \operatorname{Proj} R</math> may have multiple [[irreducible component]]s. Moreover, there may be [[nilpotent]] functions on ''X''.

Closed subschemes of <math>\mathbb{P}^n_k</math> correspond bijectively to the homogeneous ideals ''I'' of <math>k[x_0, \ldots, x_n]</math> that are [[saturated ideal|saturated]]; i.e., <math>I : (x_0, \dots, x_n) = I.</math><ref>{{harvnb|Mumford|1999|loc=pg. 111}}</ref> 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

:<math>\mathbb{P}(V) = \operatorname{Proj} k[V]</math>

where <math>k[V] = \operatorname{Sym}(V^*)</math> is the [[symmetric algebra]] of <math>V^*</math>.<ref>This definition differs from {{harvnb|Eisenbud|Harris|2000|loc=III.2.3}} but is consistent with the other parts of Wikipedia.</ref> It is the [[projectivization]] of ''V''; i.e., it parametrizes lines in ''V''. There is a canonical surjective map <math>\pi: V \setminus \{0\} \to \mathbb{P}(V)</math>, which is defined using the chart described above.<ref>cf. the proof of {{harvnb|Hartshorne|1977|loc=Ch II, Theorem 7.1}}</ref> One important use of the construction is this (cf., {{slink||Duality and linear system}}). A divisor ''D'' on a projective variety ''X'' corresponds to a line bundle ''L''. One then set

:<math>|D| = \mathbb{P}(\Gamma(X, L))</math>;

it is called the [[complete linear system]] of ''D''.

Projective space over any [[scheme (mathematics)|scheme]] ''S'' can be defined as a [[fiber product of schemes]]

:<math>\mathbb{P}^n_S = \mathbb{P}_\Z^n \times_{\operatorname{Spec}\Z} S.</math>

If <math>\mathcal{O}(1)</math> is the [[twisting sheaf of Serre]] on <math>\mathbb{P}_\Z^n</math>, we let <math>\mathcal{O}(1) </math> denote the [[pullback#Fibre-product|pullback]] of <math>\mathcal{O}(1)</math> to <math>\mathbb{P}^n_S</math>; that is, <math>\mathcal{O}(1) = g^*(\mathcal{O}(1))</math> for the canonical map <math>g: \mathbb{P}^n_{S} \to \mathbb{P}^n_{\Z}.</math>

A scheme ''X'' → ''S'' is called '''projective''' over ''S'' if it factors as a closed immersion

:<math>X \to \mathbb{P}^n_S</math>

followed by the projection to ''S''.

A line bundle (or invertible sheaf) <math>\mathcal{L}</math> on a scheme ''X'' over ''S'' is said to be [[ample line bundle|very ample relative to]] ''S'' if there is an [[Immersion (mathematics)|immersion]] (i.e., an open immersion followed by a closed immersion)

:<math>i: X \to \mathbb{P}^n_S</math>

for some ''n'' so that <math>\mathcal{O}(1)</math> pullbacks to <math>\mathcal{L}</math>. Then a ''S''-scheme ''X'' is projective if and only if it is [[proper morphism|proper]] and there exists a very ample sheaf on ''X'' relative to ''S''. Indeed, if ''X'' is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if ''X'' is projective, then the pullback of <math>\mathcal{O}(1)</math> under the closed immersion of ''X'' into a projective space is very ample. That "projective" implies "proper" is deeper: the ''[[main theorem of elimination theory]]''.