Editing Birational geometry (section)

=== Plurigenera ===
One useful set of birational invariants are the [[Kodaira dimension#the plurigenera|plurigenera]]. The [[canonical bundle]] of a smooth variety ''X'' of dimension ''n'' means the [[line bundle]] of ''n''-forms {{nowrap|1=''K<sub>X</sub>'' = Ω<sup>''n''</sup>}}, which is the ''n''th [[exterior power]] of the [[cotangent bundle]] of ''X''. For an integer ''d'', the ''d''th tensor power of ''K<sub>X</sub>'' is again a line bundle. For {{nowrap|''d'' ≥ 0}}, the vector space of global sections {{nowrap|''H''<sup>0</sup>(''X'', ''K''<sub>''X''</sub><sup>''d''</sup>)}} has the remarkable property that a birational map {{nowrap|''f'' : ''X'' ⇢ ''Y''}} between smooth projective varieties induces an isomorphism {{nowrap|''H''<sup>0</sup>(''X'', ''K''<sub>''X''</sub><sup>''d''</sup>) ≅ ''H''<sup>0</sup>(''Y'', ''K''<sub>''Y''</sub><sup>''d''</sup>)}}.{{sfn|Hartshorne|1977| loc= Exercise II.8.8.}}

For {{nowrap|''d'' ≥ 0}}, define the ''d''th '''plurigenus''' ''P''<sub>''d''</sub> as the dimension of the vector space {{nowrap|''H''<sup>0</sup>(''X'', ''K''<sub>''X''</sub><sup>''d''</sup>)}}; then the plurigenera are birational invariants for smooth projective varieties. In particular, if any plurigenus ''P''<sub>''d''</sub> with {{nowrap|''d'' > 0}} is not zero, then ''X'' is not rational.