Editing Birational geometry

{{Short description|Field of algebraic geometry}}
[[Image:Stereoprojzero.svg|thumb|right|The [[circle]] is birationally equivalent to the [[real line|line]]. One birational map between them is [[stereographic projection]], pictured here.]]

In [[mathematics]], '''birational geometry''' is a field of [[algebraic geometry]] in which the goal is to determine when two [[algebraic varieties]] are [[isomorphic]] outside lower-dimensional subsets. This amounts to studying [[Map (mathematics)|mappings]] that are given by [[rational functions]] rather than [[polynomials]]; the map may fail to be defined where the rational functions have poles.

==Birational maps==

=== Rational maps ===
A [[rational mapping|rational map]] from one variety (understood to be [[Irreducible component|irreducible]]) <math>X</math> to another variety <math>Y</math>, written as a dashed arrow {{nowrap|''X'' {{font|size=145%|⇢}}''Y''}}, is defined as a [[algebraic geometry#Morphism of affine varieties|morphism]] from a nonempty open subset <math>U \subset X</math> to <math>Y</math>. By definition of the [[Zariski topology]] used in algebraic geometry, a nonempty open subset <math>U</math> is always dense in <math>X</math>, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions.

=== Birational maps ===
A '''birational map''' from ''X'' to ''Y'' is a rational map {{nowrap|''f'' : ''X'' ⇢ ''Y''}} such that there is a rational map {{nowrap|''Y'' ⇢ ''X''}} inverse to ''f''. A birational map induces an isomorphism from a nonempty open subset of ''X'' to a nonempty open subset of ''Y'', and vice versa: an isomorphism between nonempty open subsets of ''X'', ''Y'' by definition gives a birational map {{nowrap|''f'' : ''X'' ⇢ ''Y''}}. In this case, ''X'' and ''Y'' are said to be '''birational''', or '''birationally equivalent'''. In algebraic terms, two varieties over a field ''k'' are birational if and only if their [[Function field of an algebraic variety|function fields]] are isomorphic as extension fields of ''k''.

A special case is a '''birational morphism''' {{nowrap|''f'' : ''X'' → ''Y''}}, meaning a morphism which is birational. That is, ''f'' is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of ''X'' to points in ''Y''.

=== Birational equivalence and rationality ===
A variety ''X'' is said to be '''[[Rational variety|rational]]''' if it is birational to [[affine space]] (or equivalently, to [[projective space]]) of some dimension. Rationality is a very natural property: it means that ''X'' minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset.

==== Birational equivalence of a plane conic ====
For example, the circle <math>X</math> with equation <math>x^2 + y^2 - 1 = 0</math> in the affine plane is a rational curve, because there is a rational map {{nowrap|''f'' : <math>\mathbb{A}^1</math> ⇢ ''X''}} given by
:<math>f(t) = \left( \frac{2t}{1+t^2}, \frac{1 - t^2}{1 + t^2}\right),</math>
which has a rational inverse ''g'': ''X'' ⇢ <math>\mathbb{A}^1</math> given by
:<math>g(x,y) = \frac{1-y}{x}.</math>
Applying the map ''f'' with ''t'' a [[rational number]] gives a systematic construction of [[Pythagorean triple]]s.

The rational map <math>f</math> is not defined on the locus where <math>1 + t^2 = 0</math>. So, on the complex affine line <math>\mathbb{A}^1_{\Complex}</math>, <math>f</math> is a morphism on the open subset <math>U = \mathbb{A}^1_{\Complex}-\{i, -i\}</math>, <math>f: U \to X</math>. Likewise, the rational map {{nowrap|''g'' : ''X'' ⇢ <math>\mathbb{A}^1</math>}} is not defined at the point (0,−1) in <math>X</math>.

==== Birational equivalence of smooth quadrics and P<sup>n</sup> ====
More generally, a smooth [[quadric (algebraic geometry)|quadric]] (degree 2) hypersurface ''X'' of any dimension ''n'' is rational, by [[stereographic projection]]. (For ''X'' a quadric over a field ''k'', ''X'' must be assumed to have a [[Rational point#Rational or K-rational points on algebraic varieties|''k''-rational point]]; this is automatic if ''k'' is algebraically closed.) To define stereographic projection, let ''p'' be a point in ''X''. Then a birational map from ''X'' to the projective space <math>\mathbb{P}^n</math> of lines through ''p'' is given by sending a point ''q'' in ''X'' to the line through ''p'' and ''q''. This is a birational equivalence but not an isomorphism of varieties, because it fails to be defined where {{nowrap|1=''q'' = ''p''}} (and the inverse map fails to be defined at those lines through ''p'' which are contained in ''X'').

===== Birational equivalence of quadric surface =====
The [[Segre embedding]] gives an embedding <math>\mathbb{P}^1\times\mathbb{P}^1 \to \mathbb{P}^3</math> given by
:<math>([x,y],[z,w]) \mapsto [xz,xw,yz,yw].</math>
The image is the quadric surface <math>x_0x_3=x_1x_2</math> in <math>\mathbb{P}^3</math>. That gives another proof that this quadric surface is rational, since <math>\mathbb{P}^1\times\mathbb{P}^1</math> is obviously rational, having an open subset isomorphic to <math>\mathbb{A}^2</math>.

==Minimal models and resolution of singularities==
Every algebraic variety is birational to a [[projective variety]] ([[Chow's lemma]]). So, for the purposes of birational classification, it is enough to work only with projective varieties, and this is usually the most convenient setting.

Much deeper is [[Heisuke Hironaka|Hironaka]]'s 1964 theorem on [[resolution of singularities]]: over a field of characteristic 0 (such as the complex numbers), every variety is birational to a [[Singular point of an algebraic variety|smooth]] projective variety. Given that, it is enough to classify smooth projective varieties up to birational equivalence.

In dimension 1, if two smooth projective curves are birational, then they are isomorphic. But that fails in dimension at least 2, by the [[blowing up]] construction. By blowing up, every smooth projective variety of dimension at least 2 is birational to infinitely many "bigger" varieties, for example with bigger [[Betti number]]s.

This leads to the idea of [[minimal model program|minimal models]]: is there a unique simplest variety in each birational equivalence
class? The modern definition is that a projective variety ''X'' is '''minimal''' if the [[canonical bundle|canonical line bundle]] ''K<sub>X</sub>'' has nonnegative degree on every curve in ''X''; in other words, ''K<sub>X</sub>'' is [[nef line bundle|nef]]. It is easy to check that blown-up varieties are never minimal.

This notion works perfectly for algebraic surfaces (varieties of dimension 2). In modern terms, one central result of the [[Italian school of algebraic geometry]] from 1890–1910, part of the [[Enriques–Kodaira classification|classification of surfaces]], is that every surface ''X'' is birational either to a product <math>\mathbb{P}^1\times C</math> for some curve ''C'' or to a minimal surface ''Y''.{{sfn|Kollár|Mori|1998|loc=Theorem 1.29.}} The two cases are mutually exclusive, and ''Y'' is unique if it exists. When ''Y'' exists, it is called the [[minimal model program|minimal model]] of&nbsp;''X''.

==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.

==Minimal models in higher dimensions==
{{Main|Minimal model program}}
A projective variety ''X'' is called '''minimal''' if the [[canonical bundle]] ''K<sub>X</sub>'' is [[nef line bundle|nef]]. For ''X'' of dimension 2, it is enough to consider smooth varieties in this definition. In dimensions at least 3, minimal varieties must be allowed to have certain mild singularities, for which ''K<sub>X</sub>'' is still well-behaved; these are called [[canonical singularities|terminal singularities]].

That being said, the [[minimal model program|minimal model conjecture]] would imply that every variety ''X'' is either covered by [[rational curve]]s or birational to a minimal variety ''Y''. When it exists, ''Y'' is called a '''minimal model''' of ''X''.

Minimal models are not unique in dimensions at least 3, but any two minimal varieties which are birational are very close. For example, they are isomorphic outside subsets of codimension at least 2, and more precisely they are related by a sequence of [[flip (mathematics)|flops]]. So the minimal model conjecture would give strong information about the birational classification of algebraic varieties.

The conjecture was proved in dimension 3 by Mori.{{sfn|Mori|1988}} There has been great progress in higher dimensions, although the general problem remains open. In particular, Birkar, Cascini, Hacon, and McKernan (2010){{sfn|Birkar|Cascini|Hacon|McKernan|2010}} proved that every variety of [[Kodaira dimension|general type]] over a field of characteristic zero has a minimal model.

==Uniruled varieties==
{{Main|Ruled variety}}
A variety is called '''uniruled''' if it is covered by rational curves. A uniruled variety does not have a minimal model, but there is a good substitute: Birkar, Cascini, Hacon, and McKernan showed that every uniruled variety over a field of characteristic zero is birational to a [[minimal model program|Fano fiber space]].{{efn|{{harvtxt|Birkar|Cascini|Hacon|McKernan|2010|loc=Corollary 1.3.3}}, implies that every uniruled variety in characteristic zero is birational to a Fano fiber space, using the easier result that a uniruled variety ''X'' is covered by a family of curves on which ''K<sub>X</sub>'' has negative degree. A reference for the latter fact is {{harvtxt|Debarre|2001|loc=Corollary 4.11}} and Example 4.7(1).}} This leads to the problem of the birational classification of Fano fiber spaces and (as the most interesting special case) [[Fano variety|Fano varieties]]. By definition, a projective variety ''X'' is '''Fano''' if the anticanonical bundle <math>K_X^*</math> is [[ample line bundle|ample]]. Fano varieties can be considered the algebraic varieties which are most similar to projective space.

In dimension 2, every Fano variety (known as a [[Del Pezzo surface]]) over an [[algebraically closed field]] is rational. A major discovery in the 1970s was that starting in dimension 3, there are many Fano varieties which are not [[rational variety|rational]]. In particular, smooth cubic 3-folds are not rational by [[#CITEREFClemensGriffiths1972|Clemens–Griffiths (1972)]], and smooth quartic 3-folds are not rational by [[#CITEREFIskovskihManin1971|Iskovskikh–Manin (1971)]]. Nonetheless, the problem of determining exactly which Fano varieties are rational is far from solved. For example, it is not known whether there is any smooth cubic hypersurface in <math>\mathbb{P}^{n+1}</math> with {{nowrap|''n'' ≥ 4}} which is not rational.

==Birational automorphism groups==
Algebraic varieties differ widely in how many birational automorphisms they have. Every variety of [[Kodaira dimension|general type]] is extremely rigid, in the sense that its birational automorphism group is finite. At the other extreme, the birational automorphism group of projective space <math>\mathbb{P}^n</math> over a field ''k'', known as the [[Cremona group]] ''Cr''<sub>''n''</sub>(''k''), is large (in a sense, infinite-dimensional) for {{nowrap|''n'' ≥ 2}}. For {{nowrap|1=''n'' = 2}}, the complex Cremona group <math>Cr_2(\Complex)</math> is generated by the "quadratic transformation"

: [''x'',''y'',''z''] ↦ [1/''x'', 1/''y'', 1/''z'']

together with the group <math>PGL(3,\Complex)</math> of automorphisms of <math>\mathbb{P}^2,</math> by [[Max Noether]] and [[Guido Castelnuovo|Castelnuovo]]. By contrast, the Cremona group in dimensions {{nowrap|''n'' ≥ 3}} is very much a mystery: no explicit set of generators is known.

[[#CITEREFIskovskihManin1971|Iskovskikh–Manin (1971)]] showed that the birational automorphism group of a smooth quartic 3-fold is equal to its automorphism group, which is finite. In this sense, quartic 3-folds are far from being rational, since the birational automorphism group of a [[rational variety]] is enormous. This phenomenon of "birational rigidity" has since been discovered in many other Fano fiber spaces. {{citation needed|date=April 2021}}

== Applications ==
Birational geometry has found applications in other areas of geometry, but especially in traditional problems in algebraic geometry.

Famously the minimal model program was used to construct [[moduli space]]s of varieties of general type by [[János Kollár]] and [[Nicholas Shepherd-Barron]], now known as KSB moduli spaces.{{sfn|Kollár|2013}}

Birational geometry has recently found important applications in the study of [[K-stability of Fano varieties]] through general existence results for [[Kähler–Einstein metric]]s, in the development of explicit invariants of Fano varieties to test K-stability by computing on birational models, and in the construction of moduli spaces of Fano varieties.{{sfn|Xu|2021}} Important results in birational geometry such as [[Caucher Birkar|Birkar's]] proof of boundedness of Fano varieties have been used to prove existence results for moduli spaces.

==See also==
*[[Abundance conjecture]]

==Citations==
{{reflist}}

=== Notes ===
{{notelist}}

==References==
{{refbegin|2}}
*{{Citation|ref=refAKMW | last1=Abramovich | first1=Dan | last2=Karu | first2=Kalle | last3=Matsuki | first3=Kenji | last4=Włodarczyk | first4= Jarosław | title=Torification and factorization of birational maps | journal=[[Journal of the American Mathematical Society]] | volume=15 | year=2002 | number=3 | pages=531–572 | doi=10.1090/S0894-0347-02-00396-X | mr=1896232| arxiv=math/9904135 | s2cid=18211120 }}
*{{Citation| authorlink1=Caucher Birkar | last1=Birkar | first1=Caucher | last2=Cascini | first2=Paolo | authorlink3=Christopher Hacon | last3= Hacon | first3=Christopher D. | authorlink4=James McKernan | last4=McKernan | first4=James | title=Existence of minimal models for varieties of log general type | arxiv=math.AG/0610203 | doi=10.1090/S0894-0347-09-00649-3 | mr=2601039 | year=2010 | journal=[[Journal of the American Mathematical Society]] | volume=23 | issue=2 | pages=405–468| bibcode=2010JAMS...23..405B | s2cid=3342362 }}<!--{{sfn|Birkar|Cascini|Hacon|McKernan|2010}}-->
*{{Citation|author1-last=Clemens | author1-first=C. Herbert | author1-link=Herbert Clemens | author2-link=Phillip Griffiths | author2-last=Griffiths | author2-first=Phillip A. | title=The intermediate Jacobian of the cubic threefold |mr=0302652 | year=1972 | journal=[[Annals of Mathematics]] |series= Second Series | issn=0003-486X | volume=95 | pages=281–356 | doi=10.2307/1970801 | issue=2 | jstor=1970801| citeseerx=10.1.1.401.4550 }}
*{{cite book|author1-last=Debarre | author1-first=Olivier | author1-link=Olivier Debarre | title=Higher-Dimensional Algebraic Geometry | publisher=[[Springer-Verlag]] | year=2001 | isbn=978-0-387-95227-7 | mr=1841091 }}
*{{cite book|authorlink1=Phillip Griffiths | last1=Griffiths | first1= Phillip | authorlink2=Joe Harris (mathematician) | last2= Harris | first2= Joseph | title=Principles of Algebraic Geometry | publisher=John Wiley & Sons | year=1978 | isbn=978-0-471-32792-9 | mr=0507725}}
*{{cite book|authorlink1=Robin Hartshorne | last1=Hartshorne| first1= Robin | title=Algebraic Geometry | publisher=Springer-Verlag | year=1977 | isbn= 978-0-387-90244-9 | mr=0463157}}
*{{cite book |title=Handbook of moduli|volume=2|last1=Kollár |first1=János |chapter=Moduli of varieties of general type|pages=131–157|year=2013| isbn=9781571462589|zbl=1322.14006|arxiv=1008.0621}}
*{{Citation|last1=Iskovskih | first1=V. A. | authorlink2=Yuri Manin | last2=Manin | first2=Ju. I. | title= Three-dimensional quartics and counterexamples to the Lüroth problem | doi=10.1070/SM1971v015n01ABEH001536 | mr=0291172 | year=1971 | journal=[[Matematicheskii Sbornik]] |series= Novaya Seriya | volume=86 | issue=1 | pages=140–166| bibcode=1971SbMat..15..141I }}
*{{Citation|authorlink1=János Kollár | last1=Kollár | first1=János | authorlink2=Shigefumi Mori | last2=Mori | first2=Shigefumi | title=Birational Geometry of Algebraic Varieties | publisher=[[Cambridge University Press]] | year= 1998 | isbn=978-0-521-63277-5 | mr=1658959 | doi= 10.1017/CBO9780511662560}}
*{{Citation|authorlink1=Shigefumi Mori | last1=Mori | first1=Shigefumi | title=Flip theorem and the existence of minimal models for 3-folds | jstor= 1990969 |mr=924704 | year=1988 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=1 | issue=1 | pages=117–253 | doi= 10.2307/1990969}}
*{{cite journal |doi=10.4171/EMSS/51|title=K-stability of Fano varieties: An algebro-geometric approach |year=2021 |last1=Xu |first1=Chenyang |journal=EMS Surveys in Mathematical Sciences |volume=8 |pages=265–354 |s2cid=204829174 |doi-access=free |arxiv=2011.10477 }}
{{refend}}

{{DEFAULTSORT:Birational Geometry}}
[[Category:Birational geometry| ]]