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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m1302301.png" /> be a projective morphism of algebraic varieties over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m1302302.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m1302303.png" /> (cf. also [[Algebraic variety|Algebraic variety]]). A relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m1302306.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m1302307.png" />-cycle is a formal linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m1302308.png" /> of a finite number of curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m1302309.png" /> (reduced irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023010.png" />-dimensional closed subschemes) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023011.png" /> with real number coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023013.png" /> are points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023014.png" />. (If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023015.png" />, then the word "relative" is dropped.) Two relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023016.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023017.png" />-cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023019.png" /> are said to be numerically equivalent if their intersection numbers are equal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023022.png" /> for any Cartier divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023024.png" /> (cf. also [[Divisor|Divisor]]; [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023025.png" /> of all the equivalence classes of relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023026.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023027.png" />-cycles with respect to the numerical equivalence becomes a finite-dimensional real [[Vector space|vector space]]. The closed cone of curves (the Kleiman–Mori cone) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023028.png" /> is defined to be the closed convex cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023029.png" /> generated by the classes of curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023030.png" /> which are mapped to points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023031.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023032.png" />. A half-line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023033.png" /> is called an extremal ray if the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023034.png" /> holds and if the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023036.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023037.png" />.
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Let $f : X \rightarrow S$ be a projective morphism of algebraic varieties over a field $k$ of characteristic $0$ (cf. also [[Algebraic variety|Algebraic variety]]). A relative $\mathbf{R}$-$1$-cycle is a formal linear combination $Z = \sum _ { i = 1 } ^ { t } r _ { j } C _ { j }$ of a finite number of curves $C_{j}$ (reduced irreducible $1$-dimensional closed subschemes) on $X$ with real number coefficients $r _ { j }$ such that $f ( C _ { j } )$ are points on $S$. (If $S = \operatorname{Spec} k$, then the word "relative" is dropped.) Two relative $\mathbf{R}$-$1$-cycles $Z _ { 1 }$ and $Z_2$ are said to be numerically equivalent if their intersection numbers are equal, $( D . Z _ { 1 } ) = ( D . Z _ { 2 } ) \in \bf R$ for any Cartier divisor $D$ on $X$ (cf. also [[Divisor|Divisor]]; [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). The set $N _ { 1 } ( X / S )$ of all the equivalence classes of relative $\mathbf{R}$-$1$-cycles with respect to the numerical equivalence becomes a finite-dimensional real [[Vector space|vector space]]. The closed cone of curves (the Kleiman–Mori cone) $\overline { N E } ( X / S )$ is defined to be the closed convex cone in $N _ { 1 } ( X / S )$ generated by the classes of curves on $X$ which are mapped to points on $S$ by $f$. A half-line $R = {\bf R} _ { \geq 0 } v \subset \overline { N E } ( X / S )$ is called an extremal ray if the inequality $( ( K_{X} + B ) \cdot v ) < 0$ holds and if the equality $v = v _ { 1 } + v _ { 2 }$ for $v _ { 1 } , v _ { 2 } \in \overline { NE } ( X / S )$ implies $v _ { 1 } , v _ { 2 } \in R$.
  
 
===Cone theorem.===
 
===Cone theorem.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023038.png" /> be a normal algebraic variety and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023039.png" /> an effective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023040.png" />-divisor such that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023041.png" /> is weakly log terminal (cf. [[Kawamata rationality theorem|Kawamata rationality theorem]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023042.png" /> be a projective morphism to another algebraic variety. Then there exist at most countably many extremal rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023043.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023044.png" />) satisfying the following conditions:
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Let $X$ be a normal algebraic variety and $B$ an effective $\mathbf{Q}$-divisor such that the pair $( X , B )$ is weakly log terminal (cf. [[Kawamata rationality theorem|Kawamata rationality theorem]]). Let $f : X \rightarrow S$ be a projective morphism to another algebraic variety. Then there exist at most countably many extremal rays $R _ { j } = {\bf R} _ { \geq 0 } v_j$ ($j \in J$) satisfying the following conditions:
  
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023045.png" />, there exist an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023046.png" /> and numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023047.png" />, which are zero except for finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023048.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023050.png" />.
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For any $v \in \overline { N E } ( X / S )$, there exist an element $v ^ { \prime } \in \overline { N E } ( X / S )$ and numbers $r _ { j } \in {\bf R} _ { \geq 0 }$, which are zero except for finitely many $j$, such that $( ( K _ { X } + B ) \cdot v ^ { \prime } ) \geq 0$ and $v = v ^ { \prime } + \sum_j r_j v_j$.
  
(discreteness) For any closed convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023053.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023054.png" />, there exist only finitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023056.png" />.
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(discreteness) For any closed convex cone $\Sigma$ in $N _ { 1 } ( X / S )$ such that $( ( K_{X} + B ) \cdot v ) < 0$ for any $v \in \Sigma \backslash \{ 0 \}$, there exist only finitely many $j \in J$ such that $v _ { j } \in \Sigma$.
  
 
===Contraction theorem.===
 
===Contraction theorem.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023057.png" /> be an extremal ray as above. Then there exists a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023058.png" />, called a contraction morphism, to a normal algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023059.png" /> with a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023060.png" /> which is characterized by the following properties:
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Let $R$ be an extremal ray as above. Then there exists a morphism $\phi : X \rightarrow Y$, called a contraction morphism, to a normal algebraic variety $Y$ with a morphism $g : Y \rightarrow S$ which is characterized by the following properties:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023061.png" />;
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$g \circ \phi = f$;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023062.png" />;
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$\phi_{*} {\cal O} _ { X } = {\cal O} _ { Y }$;
  
any curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023063.png" /> which is mapped to a point by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023064.png" /> is mapped to a point by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023065.png" /> if and only if its numerical class belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023066.png" />.
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any curve $C$ which is mapped to a point by $f$ is mapped to a point by $\phi$ if and only if its numerical class belongs to $R$.
  
Two methods of proofs for the cone theorem are known. The first one [[#References|[a6]]] uses a deformation theory of morphisms over a field of positive characteristic and applies only in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023067.png" /> is smooth. It is important to note that this is the only known method in mathematics to prove the existence of rational curves (as of 2000). The second approach [[#References|[a2]]] uses a vanishing theorem of cohomology groups (cf. [[Kawamata–Viehweg vanishing theorem|Kawamata–Viehweg vanishing theorem]]) which is true only in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023068.png" />. This method of proof, which is obtained via a rationality theorem (cf. [[Kawamata rationality theorem|Kawamata rationality theorem]]), applies also to singular varieties and easily extends to the logarithmic version as explained above. The contraction theorem has been proved only by a characteristic-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023069.png" /> method (cf. [[#References|[a1]]]).
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Two methods of proofs for the cone theorem are known. The first one [[#References|[a6]]] uses a deformation theory of morphisms over a field of positive characteristic and applies only in the case where $X$ is smooth. It is important to note that this is the only known method in mathematics to prove the existence of rational curves (as of 2000). The second approach [[#References|[a2]]] uses a vanishing theorem of cohomology groups (cf. [[Kawamata–Viehweg vanishing theorem|Kawamata–Viehweg vanishing theorem]]) which is true only in characteristic $0$. This method of proof, which is obtained via a rationality theorem (cf. [[Kawamata rationality theorem|Kawamata rationality theorem]]), applies also to singular varieties and easily extends to the logarithmic version as explained above. The contraction theorem has been proved only by a characteristic-$0$ method (cf. [[#References|[a1]]]).
  
In the following it is also assumed that the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023070.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023072.png" />-factorial, that is, for any prime divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023074.png" /> there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023075.png" />, depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023076.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023077.png" /> is a Cartier divisor. Then the contraction morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023078.png" /> is of one of the following types:
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In the following it is also assumed that the variety $X$ is $\mathbf{Q}$-factorial, that is, for any prime divisor $D$ on $X$ there exists a positive integer $m$, depending on $D$, such that $m D$ is a Cartier divisor. Then the contraction morphism $\phi$ is of one of the following types:
  
(Fano–Mori fibre space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023079.png" />.
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(Fano–Mori fibre space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023079.png"/>.
  
(divisorial contraction) There exists a prime divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023080.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023081.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023083.png" /> induces an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023084.png" />.
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(divisorial contraction) There exists a prime divisor $E$ of $X$ such that $\operatorname{codim}\phi ( E ) \geq 2$ and $\phi$ induces an isomorphism $X\backslash E \rightarrow Y \backslash \phi ( E )$.
  
(small contraction) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023085.png" /> is an isomorphism in codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023087.png" />, in the sense that there exists a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023088.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023089.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023090.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023091.png" /> induces an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023092.png" />.
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(small contraction) $\phi$ is an isomorphism in codimension $1$, in the sense that there exists a closed subset $E$ of codimension $\geq 2$ of $X$ such that $\phi$ induces an isomorphism $X\backslash E \rightarrow Y \backslash \phi ( E )$.
  
 
===Flip conjectures.===
 
===Flip conjectures.===
The first flip conjecture is as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023093.png" /> be a small contraction. Then there exists a birational morphism from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023094.png" />-factorial normal algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023095.png" /> which is again an isomorphism in codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023096.png" /> and is such that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023097.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023098.png" /> is weakly log terminal and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023099.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230100.png" />-ample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230101.png" />-divisor (cf. also [[Divisor|Divisor]]). The diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230102.png" /> is called a flip (or log flip). Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230103.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230104.png" />-ample.
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The first flip conjecture is as follows: Let $\phi : X \rightarrow Y$ be a small contraction. Then there exists a birational morphism from a $\mathbf{Q}$-factorial normal algebraic variety $\phi ^ { + } : X ^ { + } \rightarrow Y$ which is again an isomorphism in codimension $1$ and is such that the pair $( X ^ { + } , B ^ { + } )$ with $B ^ { + } = ( \phi _ { * } ^ { + } ) ^ { - 1 } \phi_{ *} B$ is weakly log terminal and $K _ { X ^{+}} + B ^ { + }$ is a $\phi ^ { + }$-ample $\mathbf{Q}$-divisor (cf. also [[Divisor|Divisor]]). The diagram $X \rightarrow Y \leftarrow X ^ { + }$ is called a flip (or log flip). Note that $- ( K _ { X } + B )$ is $\phi$-ample.
  
 
The second flip conjecture states that there does not exist an infinite sequence of consecutive flips.
 
The second flip conjecture states that there does not exist an infinite sequence of consecutive flips.
  
There is no small contraction if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230105.png" />. The flip conjectures have been proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230106.png" /> (see [[#References|[a3]]], [[#References|[a4]]] for the first flip conjecture, and [[#References|[a5]]], [[#References|[a7]]] for the second). The proofs depend on the classification of singularities and it is hard to extend them to a higher-dimensional case.
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There is no small contraction if $\operatorname{dim} X \leq 2$. The flip conjectures have been proved for $\operatorname { dim } X = 3$ (see [[#References|[a3]]], [[#References|[a4]]] for the first flip conjecture, and [[#References|[a5]]], [[#References|[a7]]] for the second). The proofs depend on the classification of singularities and it is hard to extend them to a higher-dimensional case.
  
 
===Minimal model program (MMP).===
 
===Minimal model program (MMP).===
Fix a base variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230107.png" /> and consider a [[Category|category]] whose objects are a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230108.png" /> and a projective morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230109.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230110.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230111.png" />-factorial normal algebraic variety and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230112.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230113.png" />-divisor such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230114.png" /> is weakly log terminal. A morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230115.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230116.png" /> in this category is a [[Birational mapping|birational mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230117.png" /> which is surjective in codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230119.png" />, in the sense that any prime divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230120.png" /> is the image of a prime divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230121.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230123.png" />. The minimal model program is a program which works under the assumption that the flip conjectures hold. It starts from an arbitrary object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230124.png" /> and constructs a morphism to another object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230125.png" /> such that one of the following holds:
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Fix a base variety $S$ and consider a [[Category|category]] whose objects are a pair $( X , B )$ and a projective morphism $f : X \rightarrow S$ such that $X$ is a $\mathbf{Q}$-factorial normal algebraic variety and $B$ is a $\mathbf{Q}$-divisor such that $( X , B )$ is weakly log terminal. A morphism from $( ( X , B ) , f )$ to $( ( X ^ { \prime } , B ^ { \prime } ) , f ^ { \prime } )$ in this category is a [[Birational mapping|birational mapping]] $\alpha : X _ { .. } \rightarrow X ^ { \prime }$ which is surjective in codimension $1$, in the sense that any prime divisor on $X ^ { \prime }$ is the image of a prime divisor on $X$, and such that $B ^ { \prime } = \alpha_{*} B$ and $f ^ { \prime } \circ \alpha = f$. The minimal model program is a program which works under the assumption that the flip conjectures hold. It starts from an arbitrary object $( ( X , B ) , f )$ and constructs a morphism to another object $( ( X ^ { \prime } , B ^ { \prime } ) , f ^ { \prime } )$ such that one of the following holds:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230126.png" /> has a Fano–Mori fibre space structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230127.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230128.png" />.
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$X ^ { \prime }$ has a Fano–Mori fibre space structure $\phi : X ^ { \prime } \rightarrow Y$ over $S$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230129.png" /> is minimal over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230130.png" /> in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230131.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230133.png" />-nef, i.e., an inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230134.png" /> holds for any curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230135.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230136.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230137.png" /> is a point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230138.png" />. Construct objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230139.png" /> inductively as follows. Set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230140.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230141.png" /> has already been constructed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230142.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230143.png" />-nef, then a minimal model is obtained. If not, then, by the cone theorem, there exists an extremal ray and one obtains a contraction morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230144.png" /> by the contraction theorem. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230145.png" />, then a Fano–Mori fibre space is obtained. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230146.png" /> is a divisorial contraction, then one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230147.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230148.png" /> is a small contraction and if the first flip conjecture is true, then take the flip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230149.png" /> and set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230150.png" />. If the second flip conjecture is true, then this process stops after a finite number of steps.
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$X ^ { \prime }$ is minimal over $S$ in the sense that $K _ { X ^ { \prime } } + B ^ { \prime }$ is $f ^ { \prime }$-nef, i.e., an inequality $( ( K _ { X ^ { \prime } } + B ^ { \prime } ) . C ) \geq 0$ holds for any curve $C$ on $X ^ { \prime }$ such that $f ( C )$ is a point on $S$. Construct objects $( ( X _ { n } , B _ { n } ) , f _ { n } )$ inductively as follows. Set $( ( X _ { 0 } , B _ { 0 } ) , f _ { 0 } ) = ( ( X , B ) , f )$. Suppose that $( ( X _ { n } , B _ { n } ) , f _ { n } )$ has already been constructed. If $K _ { X _ { n } } + B _ { n }$ is $f _ { n }$-nef, then a minimal model is obtained. If not, then, by the cone theorem, there exists an extremal ray and one obtains a contraction morphism $\phi : X _ { n } \rightarrow Y$ by the contraction theorem. If $\operatorname { dim } Y < \operatorname { dim } X _ { n }$, then a Fano–Mori fibre space is obtained. If $\phi$ is a divisorial contraction, then one sets $( ( X _ { n  + 1} , B _ { n  + 1} ) , f _ { n + 1 } ) = ( ( Y , \phi_{ * } B _ { n } ) , f _ { n } \circ \phi ^ { - 1 } )$. If $\phi$ is a small contraction and if the first flip conjecture is true, then take the flip $\phi ^ { + } : X _ { n } ^ { + } \rightarrow Y$ and set $( ( X _ { n + 1 }  , B _ { n + 1 } ) , f _ { n + 1 } ) = ( ( X _ { n } ^ { + } , ( \phi _ { * } ^ { + } ) ^ { - 1 } \phi _ { * } B _ { n } ) , f _ { n } \circ \phi ^ { - 1 } \circ \phi ^ { + } )$. If the second flip conjecture is true, then this process stops after a finite number of steps.
  
A normal algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230151.png" /> is said to be terminal, or it is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230152.png" /> has only terminal singularities, if the following conditions are satisfied:
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A normal algebraic variety $X$ is said to be terminal, or it is said that $X$ has only terminal singularities, if the following conditions are satisfied:
  
1) The canonical divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230153.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230154.png" />-Cartier divisor.
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1) The canonical divisor $K _ { X }$ is a $\mathbf{Q}$-Cartier divisor.
  
2) There exists a projective birational morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230155.png" /> from a smooth variety with a normal crossing divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230156.png" /> such that one can write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230157.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230158.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230159.png" />.
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2) There exists a projective birational morphism $\mu : Y \rightarrow X$ from a smooth variety with a normal crossing divisor $D = \sum _ { k = 1 } ^ { s } D _ { k }$ such that one can write $\mu ^ { * } K _ { X } = K _ { Y } + \sum _ { k } d _ { k } D _ { k }$ with $d _ { k } < 0$ for all $k$.
  
As a special case of the minimal model program, if one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230160.png" /> has only terminal singularities and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230161.png" />, then any subsequent pair satisfies the same condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230162.png" /> has only terminal singularities and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m130230163.png" />. This is the "non-log" version.
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As a special case of the minimal model program, if one assumes that $X$ has only terminal singularities and $B = 0$, then any subsequent pair satisfies the same condition that $X_n$ has only terminal singularities and $B _ { n } = 0$. This is the "non-log" version.
  
 
It is expected that the minimal model program works also over a field of arbitrary characteristic, although the cone and contraction theorems are conjectural in general.
 
It is expected that the minimal model program works also over a field of arbitrary characteristic, although the cone and contraction theorems are conjectural in general.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" ''Adv. Stud. Pure Math.'' , '''10''' (1987) pp. 283–360 {{MR|0946243}} {{ZBL|0672.14006}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Kawamata, "The cone of curves of algebraic varieties" ''Ann. of Math.'' , '''119''' (1984) pp. 603–633 {{MR|0750714}} {{MR|0744865}} {{ZBL|0544.14009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Mori, "Flip theorem and the existence of minimal models for 3-folds" ''J. Amer. Math. Soc.'' , '''1''' (1988) pp. 117–253 {{MR|0924704}} {{ZBL|0649.14023}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V. Shokurov, "3-fold log flips" ''Izv. Russian Akad. Nauk.'' , '''56''' (1992) pp. 105–203 {{MR|}} {{ZBL|0785.14023}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> V. Shokurov, "The nonvanishing theorem" ''Izv. Akad. Nauk. SSSR'' , '''49''' (1985) pp. 635–651 {{MR|794958}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S. Mori, "Threefolds whose canonical bundles are not numerically effective" ''Ann. of Math.'' , '''116''' (1982) pp. 133–176 {{MR|}} {{ZBL|0557.14021}} {{ZBL|0493.14020}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> Y. Kawamata, "Termination of log-flips for algebraic 3-folds" ''Internat. J. Math.'' , '''3''' (1992) pp. 653–659 {{MR|1189678}} {{ZBL|0814.14016}} </TD></TR></table>
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<table>
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<tr><td valign="top">[a1]</td> <td valign="top"> Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" ''Adv. Stud. Pure Math.'' , '''10''' (1987) pp. 283–360 {{MR|0946243}} {{ZBL|0672.14006}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> Y. Kawamata, "The cone of curves of algebraic varieties" ''Ann. of Math.'' , '''119''' (1984) pp. 603–633 {{MR|0750714}} {{MR|0744865}} {{ZBL|0544.14009}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> S. Mori, "Flip theorem and the existence of minimal models for 3-folds" ''J. Amer. Math. Soc.'' , '''1''' (1988) pp. 117–253 {{MR|0924704}} {{ZBL|0649.14023}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> V. Shokurov, "3-fold log flips" ''Izv. Russian Akad. Nauk.'' , '''56''' (1992) pp. 105–203 {{MR|}} {{ZBL|0785.14023}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> V. Shokurov, "The nonvanishing theorem" ''Izv. Akad. Nauk. SSSR'' , '''49''' (1985) pp. 635–651 {{MR|794958}} {{ZBL|}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> S. Mori, "Threefolds whose canonical bundles are not numerically effective" ''Ann. of Math.'' , '''116''' (1982) pp. 133–176 {{MR|}} {{ZBL|0557.14021}} {{ZBL|0493.14020}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> Y. Kawamata, "Termination of log-flips for algebraic 3-folds" ''Internat. J. Math.'' , '''3''' (1992) pp. 653–659 {{MR|1189678}} {{ZBL|0814.14016}} </td></tr>
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</table>

Latest revision as of 14:48, 3 February 2024

Let $f : X \rightarrow S$ be a projective morphism of algebraic varieties over a field $k$ of characteristic $0$ (cf. also Algebraic variety). A relative $\mathbf{R}$-$1$-cycle is a formal linear combination $Z = \sum _ { i = 1 } ^ { t } r _ { j } C _ { j }$ of a finite number of curves $C_{j}$ (reduced irreducible $1$-dimensional closed subschemes) on $X$ with real number coefficients $r _ { j }$ such that $f ( C _ { j } )$ are points on $S$. (If $S = \operatorname{Spec} k$, then the word "relative" is dropped.) Two relative $\mathbf{R}$-$1$-cycles $Z _ { 1 }$ and $Z_2$ are said to be numerically equivalent if their intersection numbers are equal, $( D . Z _ { 1 } ) = ( D . Z _ { 2 } ) \in \bf R$ for any Cartier divisor $D$ on $X$ (cf. also Divisor; Intersection index (in algebraic geometry)). The set $N _ { 1 } ( X / S )$ of all the equivalence classes of relative $\mathbf{R}$-$1$-cycles with respect to the numerical equivalence becomes a finite-dimensional real vector space. The closed cone of curves (the Kleiman–Mori cone) $\overline { N E } ( X / S )$ is defined to be the closed convex cone in $N _ { 1 } ( X / S )$ generated by the classes of curves on $X$ which are mapped to points on $S$ by $f$. A half-line $R = {\bf R} _ { \geq 0 } v \subset \overline { N E } ( X / S )$ is called an extremal ray if the inequality $( ( K_{X} + B ) \cdot v ) < 0$ holds and if the equality $v = v _ { 1 } + v _ { 2 }$ for $v _ { 1 } , v _ { 2 } \in \overline { NE } ( X / S )$ implies $v _ { 1 } , v _ { 2 } \in R$.

Cone theorem.

Let $X$ be a normal algebraic variety and $B$ an effective $\mathbf{Q}$-divisor such that the pair $( X , B )$ is weakly log terminal (cf. Kawamata rationality theorem). Let $f : X \rightarrow S$ be a projective morphism to another algebraic variety. Then there exist at most countably many extremal rays $R _ { j } = {\bf R} _ { \geq 0 } v_j$ ($j \in J$) satisfying the following conditions:

For any $v \in \overline { N E } ( X / S )$, there exist an element $v ^ { \prime } \in \overline { N E } ( X / S )$ and numbers $r _ { j } \in {\bf R} _ { \geq 0 }$, which are zero except for finitely many $j$, such that $( ( K _ { X } + B ) \cdot v ^ { \prime } ) \geq 0$ and $v = v ^ { \prime } + \sum_j r_j v_j$.

(discreteness) For any closed convex cone $\Sigma$ in $N _ { 1 } ( X / S )$ such that $( ( K_{X} + B ) \cdot v ) < 0$ for any $v \in \Sigma \backslash \{ 0 \}$, there exist only finitely many $j \in J$ such that $v _ { j } \in \Sigma$.

Contraction theorem.

Let $R$ be an extremal ray as above. Then there exists a morphism $\phi : X \rightarrow Y$, called a contraction morphism, to a normal algebraic variety $Y$ with a morphism $g : Y \rightarrow S$ which is characterized by the following properties:

$g \circ \phi = f$;

$\phi_{*} {\cal O} _ { X } = {\cal O} _ { Y }$;

any curve $C$ which is mapped to a point by $f$ is mapped to a point by $\phi$ if and only if its numerical class belongs to $R$.

Two methods of proofs for the cone theorem are known. The first one [a6] uses a deformation theory of morphisms over a field of positive characteristic and applies only in the case where $X$ is smooth. It is important to note that this is the only known method in mathematics to prove the existence of rational curves (as of 2000). The second approach [a2] uses a vanishing theorem of cohomology groups (cf. Kawamata–Viehweg vanishing theorem) which is true only in characteristic $0$. This method of proof, which is obtained via a rationality theorem (cf. Kawamata rationality theorem), applies also to singular varieties and easily extends to the logarithmic version as explained above. The contraction theorem has been proved only by a characteristic-$0$ method (cf. [a1]).

In the following it is also assumed that the variety $X$ is $\mathbf{Q}$-factorial, that is, for any prime divisor $D$ on $X$ there exists a positive integer $m$, depending on $D$, such that $m D$ is a Cartier divisor. Then the contraction morphism $\phi$ is of one of the following types:

(Fano–Mori fibre space) .

(divisorial contraction) There exists a prime divisor $E$ of $X$ such that $\operatorname{codim}\phi ( E ) \geq 2$ and $\phi$ induces an isomorphism $X\backslash E \rightarrow Y \backslash \phi ( E )$.

(small contraction) $\phi$ is an isomorphism in codimension $1$, in the sense that there exists a closed subset $E$ of codimension $\geq 2$ of $X$ such that $\phi$ induces an isomorphism $X\backslash E \rightarrow Y \backslash \phi ( E )$.

Flip conjectures.

The first flip conjecture is as follows: Let $\phi : X \rightarrow Y$ be a small contraction. Then there exists a birational morphism from a $\mathbf{Q}$-factorial normal algebraic variety $\phi ^ { + } : X ^ { + } \rightarrow Y$ which is again an isomorphism in codimension $1$ and is such that the pair $( X ^ { + } , B ^ { + } )$ with $B ^ { + } = ( \phi _ { * } ^ { + } ) ^ { - 1 } \phi_{ *} B$ is weakly log terminal and $K _ { X ^{+}} + B ^ { + }$ is a $\phi ^ { + }$-ample $\mathbf{Q}$-divisor (cf. also Divisor). The diagram $X \rightarrow Y \leftarrow X ^ { + }$ is called a flip (or log flip). Note that $- ( K _ { X } + B )$ is $\phi$-ample.

The second flip conjecture states that there does not exist an infinite sequence of consecutive flips.

There is no small contraction if $\operatorname{dim} X \leq 2$. The flip conjectures have been proved for $\operatorname { dim } X = 3$ (see [a3], [a4] for the first flip conjecture, and [a5], [a7] for the second). The proofs depend on the classification of singularities and it is hard to extend them to a higher-dimensional case.

Minimal model program (MMP).

Fix a base variety $S$ and consider a category whose objects are a pair $( X , B )$ and a projective morphism $f : X \rightarrow S$ such that $X$ is a $\mathbf{Q}$-factorial normal algebraic variety and $B$ is a $\mathbf{Q}$-divisor such that $( X , B )$ is weakly log terminal. A morphism from $( ( X , B ) , f )$ to $( ( X ^ { \prime } , B ^ { \prime } ) , f ^ { \prime } )$ in this category is a birational mapping $\alpha : X _ { .. } \rightarrow X ^ { \prime }$ which is surjective in codimension $1$, in the sense that any prime divisor on $X ^ { \prime }$ is the image of a prime divisor on $X$, and such that $B ^ { \prime } = \alpha_{*} B$ and $f ^ { \prime } \circ \alpha = f$. The minimal model program is a program which works under the assumption that the flip conjectures hold. It starts from an arbitrary object $( ( X , B ) , f )$ and constructs a morphism to another object $( ( X ^ { \prime } , B ^ { \prime } ) , f ^ { \prime } )$ such that one of the following holds:

$X ^ { \prime }$ has a Fano–Mori fibre space structure $\phi : X ^ { \prime } \rightarrow Y$ over $S$.

$X ^ { \prime }$ is minimal over $S$ in the sense that $K _ { X ^ { \prime } } + B ^ { \prime }$ is $f ^ { \prime }$-nef, i.e., an inequality $( ( K _ { X ^ { \prime } } + B ^ { \prime } ) . C ) \geq 0$ holds for any curve $C$ on $X ^ { \prime }$ such that $f ( C )$ is a point on $S$. Construct objects $( ( X _ { n } , B _ { n } ) , f _ { n } )$ inductively as follows. Set $( ( X _ { 0 } , B _ { 0 } ) , f _ { 0 } ) = ( ( X , B ) , f )$. Suppose that $( ( X _ { n } , B _ { n } ) , f _ { n } )$ has already been constructed. If $K _ { X _ { n } } + B _ { n }$ is $f _ { n }$-nef, then a minimal model is obtained. If not, then, by the cone theorem, there exists an extremal ray and one obtains a contraction morphism $\phi : X _ { n } \rightarrow Y$ by the contraction theorem. If $\operatorname { dim } Y < \operatorname { dim } X _ { n }$, then a Fano–Mori fibre space is obtained. If $\phi$ is a divisorial contraction, then one sets $( ( X _ { n + 1} , B _ { n + 1} ) , f _ { n + 1 } ) = ( ( Y , \phi_{ * } B _ { n } ) , f _ { n } \circ \phi ^ { - 1 } )$. If $\phi$ is a small contraction and if the first flip conjecture is true, then take the flip $\phi ^ { + } : X _ { n } ^ { + } \rightarrow Y$ and set $( ( X _ { n + 1 } , B _ { n + 1 } ) , f _ { n + 1 } ) = ( ( X _ { n } ^ { + } , ( \phi _ { * } ^ { + } ) ^ { - 1 } \phi _ { * } B _ { n } ) , f _ { n } \circ \phi ^ { - 1 } \circ \phi ^ { + } )$. If the second flip conjecture is true, then this process stops after a finite number of steps.

A normal algebraic variety $X$ is said to be terminal, or it is said that $X$ has only terminal singularities, if the following conditions are satisfied:

1) The canonical divisor $K _ { X }$ is a $\mathbf{Q}$-Cartier divisor.

2) There exists a projective birational morphism $\mu : Y \rightarrow X$ from a smooth variety with a normal crossing divisor $D = \sum _ { k = 1 } ^ { s } D _ { k }$ such that one can write $\mu ^ { * } K _ { X } = K _ { Y } + \sum _ { k } d _ { k } D _ { k }$ with $d _ { k } < 0$ for all $k$.

As a special case of the minimal model program, if one assumes that $X$ has only terminal singularities and $B = 0$, then any subsequent pair satisfies the same condition that $X_n$ has only terminal singularities and $B _ { n } = 0$. This is the "non-log" version.

It is expected that the minimal model program works also over a field of arbitrary characteristic, although the cone and contraction theorems are conjectural in general.

References

[a1] Y. Kawamata, K. Matsuda, K. Matsuki, "Introduction to the minimal model problem" Adv. Stud. Pure Math. , 10 (1987) pp. 283–360 MR0946243 Zbl 0672.14006
[a2] Y. Kawamata, "The cone of curves of algebraic varieties" Ann. of Math. , 119 (1984) pp. 603–633 MR0750714 MR0744865 Zbl 0544.14009
[a3] S. Mori, "Flip theorem and the existence of minimal models for 3-folds" J. Amer. Math. Soc. , 1 (1988) pp. 117–253 MR0924704 Zbl 0649.14023
[a4] V. Shokurov, "3-fold log flips" Izv. Russian Akad. Nauk. , 56 (1992) pp. 105–203 Zbl 0785.14023
[a5] V. Shokurov, "The nonvanishing theorem" Izv. Akad. Nauk. SSSR , 49 (1985) pp. 635–651 MR794958
[a6] S. Mori, "Threefolds whose canonical bundles are not numerically effective" Ann. of Math. , 116 (1982) pp. 133–176 Zbl 0557.14021 Zbl 0493.14020
[a7] Y. Kawamata, "Termination of log-flips for algebraic 3-folds" Internat. J. Math. , 3 (1992) pp. 653–659 MR1189678 Zbl 0814.14016
How to Cite This Entry:
Mori theory of extremal rays. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mori_theory_of_extremal_rays&oldid=23904
This article was adapted from an original article by Yujiro Kawamata (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article