View source for Mori theory of extremal rays

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" />.

===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:

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" />.

(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" />.

===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:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023061.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130230/m13023062.png" />;

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" />.

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]]]).

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:

(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" />.

(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" />.

===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.

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.

===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:

<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" />.

<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.

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:

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.

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" />.

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.

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====
<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</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</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</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</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</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</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</TD></TR></table>