Editing Birational geometry (section)

==Birational invariants==
{{Main|Birational invariant}}
At first, it is not clear how to show that there are any algebraic varieties which are not rational. In order to prove this, some birational invariants of algebraic varieties are needed. A '''birational invariant''' is any kind of number, ring, etc which is the same, or isomorphic, for all varieties that are birationally equivalent.

=== 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.

=== Kodaira dimension ===
{{Main|Kodaira dimension}}
A fundamental birational invariant is the [[Kodaira dimension]], which measures the growth of the plurigenera ''P''<sub>''d''</sub> as ''d'' goes to infinity. The Kodaira dimension divides all varieties of dimension ''n'' into {{nowrap|''n'' + 2}} types, with Kodaira dimension −∞, 0, 1, ..., or ''n''. This is a measure of the complexity of a variety, with projective space having Kodaira dimension −∞. The most complicated varieties are those with Kodaira dimension equal to their dimension ''n'', called varieties of [[Kodaira dimension|general type]].

=== Summands of ⊗<sup>''k''</sup>Ω<sup>1</sup> and some Hodge numbers ===
More generally, for any natural summand

:<math>E(\Omega^1) = \bigotimes^k \Omega^1</math>

of the ''r-''th tensor power of the cotangent bundle Ω<sup>1</sup> with {{nowrap|''r'' ≥ 0}}, the vector space of global sections {{nowrap|''H''<sup>0</sup>(''X'', ''E''(Ω<sup>1</sup>))}} is a birational invariant for smooth projective varieties. In particular, the [[Hodge theory|Hodge numbers]]

:<math>h^{p,0} = H^0(X,\Omega^p)</math>

are birational invariants of ''X''. (Most other Hodge numbers ''h''<sup>''p'',''q''</sup> are not birational invariants, as shown by blowing up.)

=== Fundamental group of smooth projective varieties ===
The [[fundamental group]] ''π''<sub>1</sub>(''X'') is a birational invariant for smooth complex projective varieties.

The "Weak factorization theorem", proved by Abramovich, Karu, Matsuki, and Włodarczyk [[#refAKMW|(2002)]], says that any birational map between two smooth complex projective varieties can be decomposed into finitely many blow-ups or blow-downs of smooth subvarieties. This is important to know, but it can still be very hard to determine whether two smooth projective varieties are birational.