In
algebraic geometry, the minimal model program is part of the birational classification of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. Its goal is to construct a birational model of any complex
projective variety which is as simple as possible. The subject has its origins in the classical
birational geometry
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 mappings that are given by rational ...
of surfaces studied by the
Italian school, and is currently an active research area within algebraic geometry.
Outline
The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational equivalence class, a variety which is "as simple as possible". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant finding a smooth variety
for which any birational
morphism with a smooth surface
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
.
In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety
, which for simplicity is assumed non-singular. There are two cases based on its
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''κ''. ...
,
:
[Note that the Kodaira dimension of an ''n''-dimensional variety is either or an integer in the range 0 to ''n''.]
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We want to find a variety
birational to
, and a morphism
to a projective variety
such that
with the
anticanonical class of a general fibre
being
ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of a ...
. Such a morphism is called a ''
Fano fibre space''.
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We want to find
birational to
, with the canonical class
nef. In this case,
is a ''minimal model'' for
.
The question of whether the varieties
and
appearing above are non-singular is an important one. It seems natural to hope that if we start with smooth
, then we can always find a minimal model or Fano fibre space inside the category of smooth varieties. However, this is not true, and so it becomes necessary to consider singular varieties also. The singularities that appear are called
terminal singularities.
Minimal models of surfaces
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the
contraction theorem of
Guido Castelnuovo
Guido Castelnuovo (14 August 1865 – 27 April 1952) was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also sign ...
essentially describes the process of constructing a minimal model of any surface. The theorem states that any nontrivial birational morphism
must contract a −1-curve to a smooth point, and conversely any such curve can be smoothly contracted. Here a −1-curve is a smooth rational curve ''C'' with self-intersection
Any such curve must have
which shows that if the canonical class is nef then the surface has no −1-curves.
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply
contract
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
all the −1-curves on the surface, and the resulting variety ''Y'' is either a (unique) minimal model with ''K'' nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface over a curve). In the second case, the ruled surface birational to ''X'' is not unique, though there is a unique one isomorphic to the product of the projective line and a curve. A somewhat subtle point is that even though a surface might have infinitely many -1-curves, one need only contract finitely many of them to obtain a surface with no -1-curves.
Higher-dimensional minimal models
In dimensions greater than 2, the theory becomes far more involved. In particular, there exist
smooth varieties which are not birational to any smooth variety
with
nef canonical class. The major conceptual advance of the 1970s and early 1980s was that the construction of minimal models is still feasible, provided one is careful about the types of singularities which occur. (For example, we want to decide if
is nef, so intersection numbers
must be defined. Hence, at the very least, our varieties must have
to be a
Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mu ...
for some positive integer
.)
The first key result is the
cone theorem
In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X.
Definition
Let X be a proper variety. By definition, a (real) ''1-cycle'' ...
of
Shigefumi Mori
is a Japanese mathematician, known for his work in algebraic geometry, particularly in relation to the classification of three-folds.
Career
Mori completed his Ph.D. titled "The Endomorphism Rings of Some Abelian Varieties" under Masayoshi Naga ...
, describing the structure of the cone of curves of
. Briefly, the theorem shows that starting with
, one can inductively construct a sequence of varieties
, each of which is "closer" than the previous one to having
nef. However, the process may encounter difficulties: at some point the variety
may become "too singular". The conjectural solution to this problem is the
flip, a kind of codimension-2 surgery operation on
. It is not clear that the required flips exist, nor that they always terminate (that is, that one reaches a minimal model
in finitely many steps.) showed that flips exist in the 3-dimensional case.
The existence of the more general log flips was established by
Vyacheslav Shokurov
Vyacheslav Vladimirovich Shokurov (russian: Вячеслав Владимирович Шокуров; born 18 May 1950) is a Russian mathematician best known for his research in algebraic geometry. The proof of the Noether–Enriques–Petri the ...
in dimensions three and four. This was subsequently generalized to higher dimensions by
Caucher Birkar
Caucher Birkar ( ku, کۆچەر بیرکار, lit=migrant mathematician, translit=Koçer Bîrkar; born Fereydoun Derakhshani ( fa, فریدون درخشانی); July 1978) is an Iranian peoples, Iranian Kurds, Kurdish mathematician and a profes ...
, Paolo Cascini,
Christopher Hacon
Christopher Derek Hacon (born 14 February 1970) is a mathematician with British, Italian and US nationalities. He is currently distinguished professor of mathematics at the University of Utah where he holds a Presidential Endowed Chair. His res ...
, and
James McKernan
James McKernan (born 1964) is a mathematician, and a professor of mathematics at the University of California, San Diego. He was a professor at MIT from 2007 until 2013.
Education
McKernan was educated at the Campion School, Hornchurch, and Tr ...
relying on earlier work of Shokurov and Hacon, and McKernan. They also proved several other problems including finite generation of log canonical rings and existence of minimal models for varieties of log general type.
The problem of termination of log flips in higher dimensions remains the subject of active research.
See also
*
Abundance conjecture In algebraic geometry, the abundance conjecture is a conjecture in
birational geometry, more precisely in the minimal model program,
stating that for every projective variety X with Kawamata log terminal singularities over a field k if the canonic ...
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Minimal rational surface
References
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*{{eom, id=M/m130230, title=Mori theory of extremal rays, authorlink=Yujiro Kawamata, first=Yujiro, last= Kawamata
Algebraic geometry
Birational geometry
3-folds