Editing Projective variety (section)

=== Degree ===
{{main|Degree of an algebraic variety|Hilbert series and Hilbert polynomial}}
Let <math>X \subset \mathbb{P}^N</math> be a projective variety. There are at least two equivalent ways to define the degree of ''X'' relative to its embedding. The first way is to define it as the cardinality of the finite set

:<math>\# (X \cap H_1 \cap \cdots \cap H_d)</math>

where ''d'' is the dimension of ''X'' and ''H''<sub>''i''</sub>'s are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if ''X'' is a hypersurface, then the degree of ''X'' is the degree of the homogeneous polynomial defining ''X''. The "general positions" can be made precise, for example, by [[intersection theory]]; one requires that the intersection is [[proper intersection|proper]] and that the multiplicities of irreducible components are all one.

The other definition, which is mentioned in the previous section, is that the degree of ''X'' is the leading coefficient of the [[Hilbert polynomial]] of ''X'' times (dim ''X'')!. Geometrically, this definition means that the degree of ''X'' is the multiplicity of the vertex of the affine cone over ''X''.<ref>{{harvnb|Hartshorne|1977|loc=Ch. V, Exercise 3.4. (e).}}</ref>

Let <math>V_1, \dots, V_r \subset \mathbb{P}^N</math> be closed subschemes of pure dimensions that intersect properly (they are in general position). If ''m<sub>i</sub>'' denotes the multiplicity of an irreducible component ''Z<sub>i</sub>'' in the intersection (i.e., [[intersection multiplicity]]), then the generalization of [[Bézout's theorem]] says:<ref>{{harvnb|Fulton|1998|loc=Proposition 8.4.}}</ref>

:<math>\sum_1^s m_i \deg Z_i = \prod_1^r \deg V_i.</math>

The intersection multiplicity ''m<sub>i</sub>'' can be defined as the coefficient of ''Z<sub>i</sub>'' in the intersection product <math>V_1 \cdot \cdots \cdot V_r</math> in the [[Chow ring]] of <math>\mathbb{P}^N</math>.

In particular, if <math>H \subset \mathbb{P}^N</math> is a hypersurface not containing ''X'', then

:<math>\sum_1^s m_i \deg Z_i = \deg(X) \deg(H)</math>

where ''Z<sub>i</sub>'' are the irreducible components of the [[scheme-theoretic intersection]] of ''X'' and ''H'' with multiplicity (length of the local ring) ''m<sub>i</sub>''.

A complex projective variety can be viewed as a [[compact complex manifold]]; the degree of the variety (relative to the embedding) is then the volume of the variety as a manifold with respect to the metric inherited from the ambient [[complex projective space]]. A complex projective variety can be characterized as a minimizer of the volume (in a sense).