In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, flips and flops are codimension-2
surgery
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...
operations arising in the
minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its or ...
, given by
blowing up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with th ...
along a
relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions.
The minimal model program
The minimal model program can be summarised very briefly as follows: given a variety
, we construct a sequence of
contractions , each of which contracts some curves on which the canonical divisor
is negative. Eventually,
should become
nef (at least in the case of nonnegative
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
), which is the desired result. The major technical problem is that, at some stage, the variety
may become 'too singular', in the sense that the canonical divisor
is no longer 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 ...
, so the intersection number
with a curve
is not even defined.
The (conjectural) solution to this problem is the ''flip''. Given a problematic
as above, the flip of
is a birational map (in fact an isomorphism in codimension 1)
to a variety whose singularities are 'better' than those of
. So we can put
, and continue the process.
[More precisely, there is a conjecture stating that every sequence ⇢ ⇢ ⇢ ⇢ of flips of varieties with Kawamata log terminal singularities, projective over a fixed normal variety terminates after finitely many steps.]
Two major problems concerning flips are to show that they exist and to show that one cannot have an infinite sequence of flips. If both of these problems can be solved, then the minimal model program can be carried out.
The existence of flips for 3-folds was proved by . The existence of log flips, a more general kind of flip, in dimension three and four were proved by
whose work was fundamental to the solution of the existence of log flips and other problems in higher dimension.
The existence of log flips in higher dimensions has been settled by . On the other hand, the problem of termination—proving that there can be no infinite sequence of flips—is still open in dimensions greater than 3.
Definition
If
is a morphism, and ''K'' is the canonical bundle of ''X'', then the relative canonical ring of ''f'' is
:
and is a sheaf of graded algebras over the sheaf
of regular functions on ''Y''.
The blowup
:
of ''Y'' along the relative canonical ring is a morphism to ''Y''. If the relative canonical ring is finitely generated (as an algebra over
) then the morphism
is called the flip of
if
is relatively ample, and the flop of
if ''K'' is relatively trivial. (Sometimes the induced birational morphism from
to
is called a flip or flop.)
In applications,
is often a
small contraction
Small may refer to:
Science and technology
* SMALL, an ALGOL-like programming language
* Small (anatomy), the lumbar region of the back
* ''Small'' (journal), a nano-science publication
* <small>, an HTML element that defines smaller text ...
of an extremal ray, which implies several extra properties:
*The exceptional sets of both maps
and
have codimension at least 2,
*
and
only have mild singularities, such as
terminal singularities In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by . Terminal singula ...
.
*
and
are birational morphisms onto ''Y'', which is normal and projective.
*All curves in the fibers of
and
are numerically proportional.
Examples
The first example of a flop, known as the Atiyah flop, was found in .
Let ''Y'' be the zeros of
in
, and let ''V'' be the blowup of ''Y'' at the origin.
The exceptional locus of this blowup is isomorphic to
, and can be blown down to
in two different ways, giving varieties
and
. The natural birational map from
to
is the Atiyah flop.
introduced Reid's pagoda, a generalization of Atiyah's flop replacing ''Y'' by the zeros of
.
References
*
*
*
*
*
*
*
*
*
*
*
*{{Citation , last1=Shokurov , first1=Vyacheslav V. , author1-link=Vyacheslav Shokurov , title=Prelimiting flips, publisher=Proc. Steklov Inst. Math. 240 , year=2003 , pages=75–213.
Algebraic geometry
Birational geometry