Editing Projective variety (section)

=== 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>