Flip (algebraic geometry)
Encyclopedia
In algebraic geometry
, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up
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.
, 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
), 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 Cartier, 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.
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.
and is a sheaf of graded algebras over the sheaf OY of regular functions on Y.
The blowup f+
of Y along the relative canonical ring is a morphism to Y. If the relative canonical ring is finitely generated (as an algebra over OY) then the morphism f+ is called the flip of f if −K is relatively ample, and the flop of f if K is relatively trivial. (Sometimes the induced birational morphism from X to X+ is called a flip or flop.)
In applications, f is often a small contraction of an extremal ray, which implies several extra properties:
Let Y be the zeros of xy = zw in A4, and let V be the blowup of Y at the origin.
The exceptional locus of this blowup is isomorphic to P1×P1, and can be blown down to P1 in 2 different ways, giving varieties X1 and X2. The natural birational map from X1 to X2 is the Atiyah flop.
introduced Reid's pagoda, a generalization of Atiyah's flop replacing Y by the zeros of
xy = (z+wk)(z−wk).
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up
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 the projectivized tangent space at that point...
along a relative canonical ring
Relative canonical model
In mathematics, the relative canonical model of a singular variety Xis a particular canonical variety that maps to X, which simplifies the structure...
. 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 nefNumerically effective
A line bundle on an algebraic variety is said to be nef , if the degree of the restriction to any algebraic curve of the variety is non-negative.In particular, every ample line bundle is nef....
(at least in the case of nonnegative Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
), 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 Cartier, 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.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 f:X→Y is a morphism, and K is the canonical bundle of X, then the relative canonical ring of f isand is a sheaf of graded algebras over the sheaf OY of regular functions on Y.
The blowup f+
of Y along the relative canonical ring is a morphism to Y. If the relative canonical ring is finitely generated (as an algebra over OY) then the morphism f+ is called the flip of f if −K is relatively ample, and the flop of f if K is relatively trivial. (Sometimes the induced birational morphism from X to X+ is called a flip or flop.)
In applications, f is often a small contraction of an extremal ray, which implies several extra properties:
- The exceptional sets of both maps f and f+ have codimension at least 2,
- X and X+ only have mild singularities, such as terminal singularities.
- f and f+ are birational morphisms onto Y, which is normal and projective.
- All curves in the fibers of f and f+ are numerically proportional.
Examples
The first example of a flop , known as the Atiyah flop, was found in .Let Y be the zeros of xy = zw in A4, and let V be the blowup of Y at the origin.
The exceptional locus of this blowup is isomorphic to P1×P1, and can be blown down to P1 in 2 different ways, giving varieties X1 and X2. The natural birational map from X1 to X2 is the Atiyah flop.
introduced Reid's pagoda, a generalization of Atiyah's flop replacing Y by the zeros of
xy = (z+wk)(z−wk).