Editing Projective variety (section)

=== The ring of sections ===
Let ''X'' be a projective variety and ''L'' a line bundle on it. Then the graded ring

:<math>R(X, L) = \bigoplus_{n=0}^{\infty} H^0(X, L^{\otimes n})</math>

is called the [[ring of sections]] of ''L''. If ''L'' is [[ample line bundle|ample]], then Proj of this ring is ''X''. Moreover, if ''X'' is normal and ''L'' is very ample, then <math>R(X,L)</math> is the integral closure of the homogeneous coordinate ring of ''X'' determined by ''L''; i.e., <math>X \hookrightarrow \mathbb{P}^N</math> so that <math>\mathcal{O}_{\mathbb{P}^N}(1)</math> pulls-back to ''L''.<ref>{{harvnb|Hartshorne|1977|loc=Ch. II, Exercise 5.14. (a)}}</ref>

For applications, it is useful to allow for [[divisor (algebraic geometry)|divisor]]s (or <math>\Q</math>-divisors) not just line bundles; assuming ''X'' is normal, the resulting ring is then called a generalized ring of sections. If <math>K_X</math> is a [[canonical divisor]] on ''X'', then the generalized ring of sections

:<math>R(X, K_X)</math>

is called the [[canonical ring]] of ''X''. If the canonical ring is finitely generated, then Proj of the ring is called the [[canonical model]] of ''X''. The canonical ring or model can then be used to define the [[Kodaira dimension]] of ''X''.