FLIPS AND FLOPS

CHRISTOPHER D. HACON AND JAMES M^{C}KERNAN

Abstract. Flips and flops are elementary birational maps which first appear in dimension three. We give examples of how flips and flops appear in many different contexts. We describe the minimal model program and some recent progress centred around the question of termination of flips.

Contents

1. Birational Geometry  1
1.1. Curves and Surfaces  1
1.2. Strong and weak factorisation  4
1.3. Flips and Flops  5
2. Minimal model program  9
3. Local approach to termination  14
4. Global approach to termination  19
5. Local-global approach to termination  21
References  26

1. Birational Geometry

1.1. Curves and Surfaces. Before we start talking about flips perhaps it would help to understand the birational geometry of curves and surfaces. For the purposes of exposition we work over the complex numbers, and we will switch freely between the algebraic and holomorphic perspective. Example 1.1. Consider the function

ϕ: ℂ^{2} −−→ ℂ  defined by the rule  (x, y) ⟶ y/x.

Geometrically this is the function which assigns to every point (x, y) the slope of the line connecting (0, 0) to (x, y). This function is not defined where x = 0 (the slope is infinite here). One can partially remedy this situation by replacing the complex plane ℂ by the Riemann sphere ℙ^{1} = ℂ ∪ {∞}. We get a function

Date: August 26, 2010.

We would like to thank Chenyang Xu for his help with the contents of this paper.

ϕ: ℂ^{2} −−→ ℙ^{1}  defined by the rule  (x, y) ⟶ [X : Y].

However ϕ is still not defined at the origin of ℂ^{2}. Geometrically this is clear, since it does not make sense to ask for the slope of the line connecting the origin to the origin. In fact, if one imagines approaching the origin along a line through the origin then ϕ is constant along any such line and picks out the slope of this line. So it is clear that we cannot extend ϕ to the whole of ℂ^{2} continuously.

It is convenient to have some notation to handle functions which are not defined everywhere:

Definition 1.2. Let X be an irreducible quasi-projective variety and let Y be any quasi-projective variety. Consider pairs (f, U), where U ⊂ X is an open subset and f: U ⟶ Y is a morphism of quasi-projective varieties. We say two pairs (f, U) and (g, V) are equivalent if there is an open subset W ⊂ U ∩ V such that f|_{W} = g|_{W}.

A rational map ϕ: X −−→ Y is an equivalence class of pairs (f, U).

In fact if ϕ is represented by (f, U) and (g, V) then ϕ is also represented by (h, U ∪ V) where

h(x) = \begin{cases}f(x) & x ∈ U \\ g(x) & x ∈ V. \end{cases}

So there is always a largest open subset where ϕ is defined, called the domain of ϕ, denoted dom ϕ. The locus of points not in the domain of ϕ is called the indeterminacy locus.

Example 1.3. Let C be the conic in ℙ^{2} defined by the equation

X^{2} + Y^{2} = Z^{2}.

Consider the rational map

ϕ: C −−→ ℙ^{1}  defined by the rule  [X : Y : Z] ⟶ [X : Y − Z].

It would seem that ϕ is not defined where both X = 0 and Y = Z and of course X^{2} + Y^{2} = Z^{2}, that is, at the point [0 : 1 : 1].

If one passes to the open subset U = ℂ^{2}, where Z ≠ 0, and introduces coordinates x = X/Z and y = Y/Z then C_{0} = C ∩ U is defined by the equation x^{2} + y^{2} = 1 and the map above reduces to the function

C_{0} −−→ ℂ  defined by the rule  (x, y) ⟶ x/(y − 1),

which would again not seem to be defined at the point (0, 1) of the curve C_{0}.

However note that (Y − Z)(Y + Z) = Y^{2} − Z^{2} = X^{2} on the curve C. Therefore, on the open set C − {[0 : 1 : 1], [0 : −1 : 1]},

[X(Y + Z) : (Y − Z)(Y + Z)] = [X(Y + Z) : X^{2}] = [Y + Z : X].

Thus ϕ is also the function

ϕ: C −−→ ℙ^{1}  defined by the rule  [X : Y : Z] ⟶ [Y + Z : X].

It is then clear that ϕ is in fact a morphism, defined on the whole of the smooth curve C.

In fact the most basic result in birational geometry is that every map from a smooth curve to a projective variety always extends to a morphism:

Lemma 1.4. Let f: C −−→ X be a rational map from a smooth curve to a projective variety. Then f is a morphism, that is, the domain of f is the whole of C.

Proof. As X is a closed subset of ℙ^{n}, it suffices to show that the composition C −−→ ℙ^{n} is a morphism. So we might as well assume that X = ℙ^{n}. C is abstractly a Riemann surface. Working locally we might as well assume that C = Δ, the unit disk in the complex plane ℂ. We may suppose that f is defined outside of the origin and we want to extend f to a function on the whole unit disk. Let z be a coordinate on the unit disk. Then f is locally represented by a function

z ⟶ [f_{0} : f_{1} : ⋯: f_{n}],

where each f_{i} is a meromorphic function of z with a possible pole at zero. It is well known that f_{i}(z) = z^{m_{i}}g_{i}(z), where g_{i}(z) is holomorphic and does not vanish at zero and m_{0}, m_{1}, …, m_{n} are integers. Let m = min m_{i}. Then f is locally represented by the function

z ⟶ [h_{0} : h_{1} : ⋯: h_{n}],

where h_{i}(z) = z^{−m}f_{i}(z). As h_{0}, h_{1}, …, h_{n} are holomorphic functions and at least one of them does not vanish, it follows that f is a morphism.

□

Note that the birational classification of curves is easy. If two curves are smooth and birational then they are isomorphic. In particular, two curves are birational if and only if their normalisations are isomorphic.

Definition 1.5. Let ϕ: X −−→ Y be a rational map between two irreducible quasi-projective varieties. The graph of ϕ, denoted Γ_{ϕ}, is the closure in X × Y of the graph of the function f: U ⟶ Y, where ϕ is represented by the pair (f, U).

Definition 1.6. Consider the rational function

ϕ: ℂ^{2} −−→ ℂ  defined by the rule  (x, y) ⟶ y/x,

which appears in (1.1). Then the graph Γ_{ϕ} ⊂ ℂ^{2} × ℙ^{1} is the zero locus of the polynomial xT = yS, where (x, y) are coordinates on ℂ^{2} and [S : T] are homogeneous coordinates on ℙ^{1}. Consider projection onto the first factor π: Γ_{ϕ} ⟶ ℂ^{2}. Away from the origin this morphism is an isomorphism but the inverse image E of the origin is a copy of ℙ^{1}. π is called the blow up of the origin and E is called the exceptional divisor.

We note that π has a simple description in terms of toric geometry. ℂ^{2} corresponds to the cone spanned by (0, 1) and (1, 0). Γ_{ϕ} is the union of the two cones spanned by (1, 0) and (1, 1) and (1, 1) and (0, 1); it is obtained in an obvious way by inserting the vector (1, 1).

Given any smooth surface S, we can define the blow up of a point p ∈ S by using local coordinates. More generally given any smooth quasi-projective variety X and a smooth subvariety V, we may define the blow up π: Y ⟶ X of V inside X. π is a birational morphism, which is an isomorphism outside V. The inverse image of V is a divisor E; the fibres of E over V are projective spaces of dimension one less than the codimension of V in X and in fact E is a projective bundle over V. V is called the centre of E.

For example, to blow up one of the axes in ℂ^{3}, the toric picture is again quite simple. Start with the cone spanned by (1, 0, 0), (0, 1, 0) and (0, 0, 1), corresponding to ℂ^{3} and insert the vector (1, 1, 0) = (1, 0, 0) + (0, 1, 0). We get two cones one spanned by (1, 0, 0), (1, 1, 0) and (0, 0, 1) and the other spanned by (0, 1, 0), (1, 1, 0) and (0, 0, 1). To blow up the origin, insert the vector (1, 1, 1). There are then three cones. One way to encode this data a little more efficiently is to consider the triangle (two dimensional simplex) spanned by (1, 0, 0), (0, 1, 0) and (0, 0, 1) and consider the intersection of the corresponding cones with this triangle.

1.2. Strong and weak factorisation. We have the following consequence of resolution of singularities, see [16]:

Theorem 1.7 (Resolution of indeterminancy; Hironaka). Let ϕ: X −−→ Y be a rational map between two quasi-projective varieties.

If X is smooth, then there is a sequence of blow ups π: W ⟶ X along smooth centres such that the induced rational map ψ: W ⟶ Y is a morphism.

For surfaces we can do much better in the case of a birational map:

Theorem 1.8. Let ϕ: S −−→ T be a birational map between two smooth quasi-projective surfaces.

Then there is a smooth surface W and two birational morphisms π: W ⟶ S and π′: W ⟶ T, both of which are compositions of blow ups along smooth centres.

Example 1.9. Consider the function

ϕ: ℙ^{2} −−→ ℙ^{2} defined by the rule [X : Y : Z] ⟶ [X^{−1} : Y^{−1} : Z^{−1}].

Then ϕ is a birational map, an involution of ℙ^{2}. As

[X^{−1} : Y^{−1} : Z^{−1}] = [Y Z : XZ : Y Z],

it is not hard to see that ϕ sends the three coordinate axes to the coordinate points. But then it follows that the coordinate points [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1] are part of the indeterminacy locus of ϕ. In fact, if we blow up π: W ⟶ ℙ^{2} the three coordinate points, then ϕ blows down the strict transform of the three coordinate axes π′: W ⟶ ℙ^{2}.

Consider the standard fan for ℙ^{2}, the union of the three cones spanned by (1, 0), (0, 1) and (−1, −1). Blowing up the coordinate points, corresponds to inserting the three vectors (1, 1) = (1, 0) + (0, 1), (0, −1) = (1, 0) + (−1, −1) and (−1, 0) = (−1, −1) + (0, 1). The resulting fan is the fan for the toric variety W. Note that the strict transforms of the three coordinate axes are now contractible as (1, 0) = (1, 1) + (0, −1), (−1, −1) = (−1, 0) + (0, −1) and (0, 1) = (1, 1) + (−1, 0).

1.3. Flips and Flops. It is conjectured that a result similar to (1.8) holds in all dimensions:

Conjecture 1.10 (Strong factorisation). Let ϕ: X −−→ Y be a birational map between two quasi-projective varieties.

Then there is a quasi-projective variety W and two birational morphisms π: W ⟶ X and π′: W ⟶ Y which are both the composition of a sequence of blow ups of smooth centres.

Unfortunately we only know a weaker statement, see [1] and [44]:

Theorem 1.11 (Weak factorisation: Abramovich, Karu, Matsuki, Włodarczyk; Włodarczyk). Let ϕ: X −−→ Y be a birational map between two quasi-projective varieties.

Then we may factor ϕ into a sequence of birational maps ϕ_{1}, ϕ_{2}, …, ϕ_{m}, ϕ_{i}: X_{i} −−→ X_{i+1} and there are quasi-projective varieties W_{1}, W_{2}, …, W_{m} and two birational morphisms π_{i}: W_{i} ⟶ X_{i} and π′_{i}: W_{i} ⟶ X_{i+1} which are both the composition of a sequence of blow ups of smooth centres.

The problem is that beginning with threefolds there are birational maps which are isomorphisms in codimension two: Example 1.12. Suppose we start with ℂ^{3} and blow up both the x-axis and the y-axis. Suppose we first blow up the x-axis and then the y-axis to get X ⟶ ℂ^{3}. Let E_{x} be the exceptional divisor over the x-axis, with strict transform E′_{x} and let E_{y} be the exceptional divisor over the y-axis. The strict transform of the y-axis intersects E_{x} in a point. When we blow this up, we also blow up this point of E_{x}. So E′_{x} has one reducible fibre with two components and E_{y} is a ℙ^{1}-bundle over the y-axis. If we blow up Y ⟶ ℂ^{3} in the opposite order then E_{x} is now a ℙ^{1}-bundle and the strict transform E′_{y} contains one reducible fibre. The resulting birational map X −−→ Y is an isomorphism outside the extra copies of ℙ^{1} belonging to E′_{x} and E′_{y}. On the other hand it is not an isomorphism along these curves. This is the simplest example of a flop.

The language of fans and toric geometry is very convenient. We start with the cone spanned by e_{1} = (1, 0, 0), e_{2} = (0, 1, 0) and e_{3} = (0, 0, 1). Blowing up the x-axis corresponds to inserting the vector e_{2} + e_{3} and we get two cones, σ_{1}, spanned by e_{1}, e_{2} + e_{3} and e_{2} and σ_{2} spanned by e_{1}, e_{2} + e_{3} and e_{3}. Blowing up the y-axis we insert the vector e_{1} + e_{3}, so that we subdivide σ_{2} into two more cones, one spanned by e_{1}, e_{2} + e_{3} and e_{1} + e_{3} and the other spanned by e_{3}, e_{2} + e_{3}, e_{1} + e_{3}.

Now suppose that we reverse the order. At the first step we insert the vector e_{1} + e_{3} and we get two cones, τ_{1} spanned by e_{1}, e_{1} + e_{3} and e_{2} and τ_{2} spanned by e_{2}, e_{1} + e_{3} and e_{3}. At the next step we insert the vector e_{2} + e_{3}, and subdivide τ_{2} into two cones, one spanned by e_{2}, e_{1} + e_{3} and e_{2} + e_{3} and the other spanned by e_{3}, e_{1} + e_{3} and e_{2} + e_{3}.

In fact to prove (1.10) it suffices to prove it in the special case of toric varieties. For an interesting explanation of the difficulties in proving strong factorisation, see [17].

There is another way to construct this flop: Example 1.13. Let Q be the quadric cone xz − yt = 0 inside ℂ^{4}. If we blow up the origin we get a birational morphism W ⟶ Q with exceptional E divisor isomorphic ℙ^{1} ×ℙ^{1}. We can partially contract E, by picking one of the projection maps, W ⟶ X and W ⟶ Y. The resulting birational map X −−→ Y is the same as the flop introduced above.

Perhaps the easiest way to see this is to use toric geometry. Q corresponds to the cone spanned by four vectors v_{1}, v_{2}, v_{3} and v_{4} in ℝ^{3}, belonging to the standard lattice ℤ^{3}, any three of which span the standard lattice, such that v_{1} +v_{3} = v_{2} +v_{4}. W corresponds to inserting the vector v_{1} + v_{3} and subdividing the cone into four subcones. X and Y correspond to the two different ways to pair off the four maximal cones into two cones.

One particularly nice feature of the toric description is that we can modify the picture above to get lots of examples of flips and flops. Suppose we pick any four vectors v_{1}, v_{2}, v_{3} and v_{4} belonging to the standard lattice which span a strongly convex cone. Then a_{1}v_{1} + a_{3}v_{3} = a_{2}v_{2} + a_{4}v_{4}, for some positive integers a_{1}, a_{2}, a_{3} and a_{4}. Once again we can insert the vector a_{1}v_{1} + a_{3}v_{3} and pair off the resulting cones to get two different toric threefolds X and Y which are isomorphic in codimension one.

The simplest example of a flip is when 2v_{1} + v_{3} = v_{2} + v_{4}. If we start with the wall connecting v_{2} and v_{4} then the flip corresponds to replacing this by the wall connecting v_{1} and v_{3}. X has one singular point, which is a ℤ^{2}-quotient singularity, corresponding to the cone spanned by v_{2}, v_{3} and v_{4}. Indeed, 2v_{1} is an integral linear combination of these vectors but not v_{1}. On the other hand, Y is smooth.

Another place that flops appear naturally is in the example of a Cremona transformation of ℙ^{3}.

Example 1.14. Consider the function

ϕ: ℙ^{3} −−→ ℙ^{3} defined by the rule [X : Y : Z : T] ⟶ [X^{−1} : Y^{−1} : Z^{−1} : T^{−1}].

Then ϕ is a birational automorphism of ℙ^{3}. The graph of this function first blows up the four coordinate points, to get four copies of ℙ^{2}, then the six coordinate axes, to get six copies of ℙ^{1} × ℙ^{1}. The reverse map then blows down those six copies of ℙ^{1} × ℙ^{1}, but this time using the other projection and then we finally blow down the strict transforms of the four coordinate planes.

Note that if we just blow up the four coordinate points on both sides then the resulting threefolds are connected by six flops. All of this is easy to describe using toric geometry; the picture is similar to the picture above of the Cremona transformation of ℙ^{2}.

One can use flops to construct some interesting examples.

Example 1.15 (Hironaka). Suppose we start with X = ℙ^{3} and two conics C_{1} and C_{2} which intersect in two points p and q. Imagine blowing up both C_{1} and C_{2} but in a different order at p and q. Suppose we blow up first C_{1} and then C_{2} over p but first C_{2} and then C_{1} over q. Let π: M ⟶ ℙ^{3} be the resulting birational map. Note that π does not exist in the category of varieties but that this construction does make sense in the category of complex manifolds.

I claim that even the exceptional locus E_{1} ∪ E_{2} is not a projective variety. Let l be general fibre of the exceptional divisor E_{1} over C_{1}, let l_{1} + l_{2} be the reducible fibre over p, let m be the general fibre of E_{2} over C_{2} and let m_{1} +m_{2} be the reducible fibre over q. Suppose the irreducible fibre of E_{2} over p is attached to l_{1} and the irreducible fibre of E_{1} over q is attached to m_{1}. Note that

m_{1} ≡ l ≡ l_{1} + l_{2} ≡ m + l_{2} ≡ m_{1} + m_{2} + l_{2},

where ≡ denotes numerical equivalence. This implies that l_{2} + m_{2} ≡ 0. If M is projective then a hyperplane class H would intersect l_{2} + m_{2} positively, a contradiction.

Note that M is related to a projective variety Y over X by an (analytic) flop. Just flop either l_{2} or m_{2}.

Example 1.16 (Atiyah). Suppose that we start with a family of quartic surfaces in ℙ^{3} degenerating to a quartic surface with a simple node (a singularity which in local analytic coordinates resembles x^{2}+y^{2}+z^{2} = 0 in ℂ^{3}). It is a simple matter to find a degeneration whose total space has a singularity locally of the form xz − yt. In this case we can blow up this singularity in two different ways, see (1.13), to get two different families of smooth K3 surfaces, which are connected by a flop.

But now we have two distinct families of K3 surfaces, which agree outside one point. In fact even though the families are different they have isomorphic fibres. It follows that the moduli space of K3 surfaces is not Hausdorff.

Example 1.17 (Reid). Let X_{0} ⊂ ℂ^{4} be the smooth threefold given by the equation

y^{2} = ((x − a)^{2} − t_{1})((x − b)^{2} − t_{2}),

where x, y, t_{1}, t_{2} are coordinates on ℂ^{4} and a ≠ b are constants. Let X be the closure of X_{0} in ℙ^{1} × ℙ^{1} × ℂ^{2}. Projection π: X ⟶ ℂ^{2} down to ℂ^{2} with coordinates t_{1} and t_{2} realises X as a family of projective curves of genus one over ℂ^{2}. If t_{1}t_{2} ≠ 0 then we have a smooth curve of genus one, that is an elliptic curve. If t_{1} = 0 and t_{2} ≠ 0 or t_{2} = 0 and t_{1} ≠ 0 then we get a nodal rational curve (a copy of ℙ^{1} with two points identified). If t_{1} = t_{2} = 0 then we get a pair C_{1} ∪ C_{2} of copies of ℙ^{1} joined at two points.

One can check that both C_{1} and C_{2} can be contracted individually to a simple node. Therefore we can flop either C_{1} or C_{2}. Suppose that we flop C_{1}. Since C_{1} is contracted by π this flop is over S so that the resulting threefold Y admits a morphism to ψ: Y ⟶ ℂ^{2}. We haven’t changed the morphism π outside s and one can check that the fibre over (0, 0) of ψ is a union D_{1} ∪D_{2} of two copies of ℙ^{1} which intersect in two different points. Once again we can flop either of these curves. Suppose that D_{2} is the strict transform of C_{2} so that D_{1} is the flopped curve. If we flop D_{1} then we get back to X but if we flop D_{2} then we get another threefold which fibres over S. Continuing in this way we get infinitely many threefolds all of which admit a morphism to S and all of which are isomorphic over the open set S − {s}. Let G be the graph whose vertices are these threefolds, where we connect two vertices by an edge if there is a flop between the two threefolds over S. Let G′ be the graph whose vertices are the integers where we connect two vertices i and j if and only if |i − j| = 1. Then G and G′ are isomorphic.

2. Minimal model program

The idea behind the minimal model program (which we will abbreviate to MMP) is to find a particularly simple birational representative of every projective variety. For curves we have already seen that two smooth curves are birational if and only if they are isomorphic. For surfaces there are non-trivial birational maps, but by (1.8) only if there are rational curves (non-constant images of ℙ^{1}). Roughly speaking, simple means that we cannot contract any more rational curves. In practice it turns out that we don’t want to contract every curve, just those curves on which the canonical divisor is negative.

Definition 2.1. Let X be a normal projective variety. A divisor D = ∑ n_{i}D_{i} is a formal linear combination of codimension one subvarieties.

The canonical divisor K_{X} is the divisor associated to the zeroes and poles of any meromorphic differential form ω.

Note that the canonical divisor is really an equivalence class of divisors.

Example 2.2. If X = ℙ^{1} and z is the standard coordinate on ℂ then dz/z is a meromorphic differential form. It has a pole at zero and a pole at infinity, since

\frac{d(1/z)}{1/z} = −\frac{dz}{z} .

If p represents zero and q infinity then K_{ℙ^{1}} = −p − q. If we started with dz/z^{2} then K_{ℙ^{1}} = −2p (a double pole at zero) but if we start with dz then K_{ℙ^{1}} = −2q (a double pole at infinity). And so on. If X is an elliptic curve E then it is a one dimensional complex torus, the quotient of ℂ by a lattice isomorphic to ℤ^{2}. In this case the differential form dz descends to the torus (as it is translation invariant) and K_{E} = 0 (no zeroes or poles). If C has genus g ≥ 2 then deg K_{C} = 2g − 2 > 0.

For ℙ^{n} we have K_{ℙ^{n}} = −(n + 1)H, where H is the class of a hyperplane. More generally still, suppose X is a projective toric variety. Then a dense open subset of X is isomorphic to a torus (ℂ^{∗})^{n}. A natural holomorphic differential n-form which is invariant under the action of the torus is

\frac{dz_{1}}{z_{1}} ∧ \frac{dz_{2}}{z_{2}} ∧ ⋯ ∧ \frac{dz_{n}}{z_{n}} .

This form extends naturally to a meromorphic differential on the whole toric variety with simple poles along the invariant divisors. In other words,

K_{X} + Δ ∼_{ℚ} 0,

where Δ = ∑ D_{i} is a sum of the invariant divisors. In the case of ℙ^{n} there are n + 1 invariant divisors corresponding to the n + 1 coordinate hyperplanes.

One of the most useful ways to compute the canonical divisor is the adjunction formula. If M is a smooth variety and X is a smooth divisor then

(K_{M} + X)|_{X} = K_{X}.

For example, if X is a quartic surface in ℙ^{3} then

K_{X} = (K_{ℙ^{3}} + X)|_{X} = (−4H + 4H)|_{X} = 0.

Together with the fact that smooth hypersurfaces of dimension at least two are simply connected this implies that X is a K3 surface.

Suppose that T ⟶ S is the blow up of a point with exceptional divisor E ≃ ℙ^{1}. It is straightforward to check that the self-intersection E^{2} = E · E = −1. By adjunction we have

−2 = K_{ℙ^{1}} = K_{E} = (K_{T} + E)|_{E} = K_{T} · E + E^{2}.

It follows that K_{T} ·E = −1. For obvious reasons we call any such curve a −1-curve. The idea of the MMP is to only contract curves on which the canonical divisor is negative.

Definition 2.3. Let X be a normal projective variety and let D be a Cartier divisor (something locally defined by a single equation). We say that D is nef if D · C ≥ 0 for every curve C ⊂ X.

Let us first see how the minimal model program works for surfaces. Step 0: Start with a smooth surface S. Step 1: Is K_{S} nef? If yes, then stop. S is a minimal model. Step 2: If no, then there must be a curve C such that K_{S} · C <

0. We can always choose C so that there is a contraction morphism π: S ⟶ T which contracts C and there are three cases:

(i) S = ℙ^{2}, T is a point and C is a line.

(ii) T is a curve, S is a ℙ^{1}-bundle over T and C is a fibre.

(iii) T is a smooth surface, π is a blow up of a point on T and C is

the exceptional divisor. Step 3: If we are in case (i) or (ii), then stop. Otherwise replace S

by T and go back to Step 1.

The fact that we can always find a curve C to contract is a non-trivial result, due to the Italian school of algebraic geometry. It is possible that at Step 1 there is more than one choice of π.

Example 2.4. Suppose that we start with the blow up S of ℙ^{2} at two different points p and q. There are three relevant curves, E and F the exceptional divisors over p and q and L, the strict transform of the line connecting p and q.

At the first step of the K_{S}-MMP we are presented with three choices. We can choose to contract E, F or L, since all three of these curves are −1-curves. If we contract E, π: S ⟶ T, then at the next step we are presented with two choices of curves to contract on T. We can either contract the image of F, in which case the end product of the MMP is the original ℙ^{2}. On the other hand, there is a morphism T ⟶ ℙ^{1}. Every fibre is isomorphic to ℙ^{1}, L is a fibre and F is a section of this morphism. This is a possible end product of the MMP. If instead we decide to contract F, then we get almost exactly the same picture; note however that even though the two ℙ^{1}-bundles we get are isomorphic, the induced birational map between them is not an isomorphism. However if we choose to contract L then the resulting surface is isomorphic to ℙ^{1} × ℙ^{1}. Projection to either factor ℙ^{1} × ℙ^{1} ⟶ ℙ^{1} are two possible other end products of the MMP.

Once again the language of toric geometry gives a convenient way to encode this picture. S corresponds to the fan with one dimensional rays spanned by (1, 0), (0, 1), (−1, 0), (−1, −1) and (0, −1). Blowing down E and F corresponds to removing the two rays (−1, 0) and (0, −1). Blowing down E corresponds to removing (−1, 0) and the morphism to ℙ^{1} corresponds to the projection of ℝ^{2} onto the x-axis. Contracting L corresponds to removing (−1, −1); the resulting fan is clearly the fan for ℙ^{1} × ℙ^{1}.

The most important feature of any algorithm is termination. Termination for surfaces is clear. Every time we contract a copy of ℙ^{1}, topologically we are replacing a copy of the sphere S^{2} by a point. Consequently the second Betti number b_{2}(S) drops by one and so the MMP terminates after at most b_{2}(S)-steps. Equivalently the Picard number drops by one at every step.

One interesting application of the MMP for surfaces is in the construction of a compactification \overline{M}_{g} of the moduli space of curves M_{g}. In particular suppose we are given a family π: S_{0} ⟶ C_{0} of smooth projective curves over a smooth affine curve C_{0}. Then there is a unique projective curve C which contains C_{0} as an open subset. We would like to complete S_{0} to a family of curves π: S ⟶ C, which makes the following diagram commute:

S_{0} ⟶ S
π_{0}↓  ↓^{π}
C_{0} ⟶ C.

Here the horizontal arrows are inclusions. We would like the fibres of π to be nodal projective curves, whose canonical divisor is ample. The first observation is that this is not in fact possible. In general we can only fill in this family after a finite cover of C_{0}.

Here is the general algorithm. The first step is to pick any compactification of S_{0} and of the morphism π_{0}. The next step is to blow up S, so that the reduced fibres are curves with nodes. After this we take a cover of C and replace S by the normalisation of the fibre product. If the cover of C is sufficiently ramified along the singular fibres of π this step will eliminate the multiple fibres. The penultimate step is to run the MMP over C. This has the effect of contracting all −1-curves contained in the fibres of π. The final step is to contract all −2-curves, that is, all copies of ℙ^{1} with self-intersection −2.

We now consider the MMP in higher dimension. There is a similar picture, except that we also encounter flips:

Definition 2.5. Let π: X ⟶ Z be a birational morphism. We say that π is small if π does not contract a divisor. We say that π is a flipping contraction if −K_{X} is ample over Z and the relative Picard number is one. The flip of π is another small birational morphism ψ: Y ⟶ Z of relative Picard number one such that K_{Y} is ample over Z.

The relative Picard number is the difference in the Picard numbers. The relative Picard number is one if and only if every two curves contracted by π are numerically multiples of each other. In this case a ℚ-Cartier divisor D is ample over Z if and only if D · C > 0 for one curve C contracted by π.

Flops are defined similarly, except that now K_{X} and K_{Y} are trivial over Z and yet the induced birational map X −−→ Y is not an isomorphism. The MMP in higher dimensions proceeds as follows:

Step 0: Start with a smooth projective variety X.

Step 1: Is K_{X} nef? If yes, then stop. X is a minimal model.

Step 2: If no, then there must be a curve C such that K_{X} · C < 0. We can always choose C so that there is a contraction morphism π: X ⟶ Z which contracts C and there are two cases:

(i) dim Z < dim X. C is contained in a fibre. The fibres F of π are Fano varieties, so that −K_{F} is ample. π is a Mori fibre space.

(ii) dim Z = dim X. In this case π is birational and there are two sub cases:

(a) π contracts a divisor E.

(b) π is small. Step 3: If we are in case (i), then stop. If we are in case (a) then

replace X by Z and go back to (1). If we are in case (b) then replace X by the flip X −−→ Y and go back to (1).

The fact that we may find C and π at step 2 is quite subtle, and is due to the work of many people, including Kawamata, Kollár, Miyaoka, Mori, Reid, Shokurov and many others. For more details see, for example, the book by Kollár and Mori, [25]. For an excellent survey of flips and flops, especially for threefolds, see [22].

Existence of terminal 3-flips was first proved by Mori, [31]. Kollár and Mori give a complete classification of all terminal flips in [24]. Shokurov proved the existence of 4-fold flips, [40]. Existence in all dimensions was proved in [14] and [15]:

Theorem 2.6 (Existence: Hacon, M^{c}Kernan). Flips exist in all dimensions.

Actually stating things this way is a considerable simplification; we also need the main result of [6] to finish a somewhat involved induction. The proof of (2.6) draws considerable inspiration and ideas from two sources. First, Siu’s theory of multiplier ideals and his proof of deformation invariance of plurigenera, see [43], especially the recasting of these ideas in the algebraic setting [19], due to Kawamata. Second, Shokurov’s theory of saturation of the restricted algebras and his proof of the existence of flips for fourfolds, [40], all of which is succinctly explained in Corti’s book, [9].

We have already seen (2.4) that the end product of the MMP is not unique. For surfaces the minimal model is unique. If X is a threefold and X −−→ Y is a flop then X is minimal if and only if Y is minimal, so there is often more than minimal model. In fact, Kawamata [20] proved that any two minimal models are connected by a sequence of flops.

Example 2.7. Suppose we start with the elliptic fibration π: X ⟶ S given in (1.17). Possibly replacing S by a finite cover, we may assume that S contains no rational curves. Suppose that we run the K_{X}-MMP. Every step of the MMP the locus we contract is covered by rational curves. It follows that every step of the MMP is over S and the end product of the MMP is a minimal model. The MMP preserves the property that one isolated fibre is the union of two copies of ℙ^{1} meeting in two points. It follows that X has infinitely many minimal models.

Kawamata has similar examples of Calabi-Yau threefolds with infinitely many minimal models, [18].

If we get down to a Mori fibre space the situation is considerably more complicated, as (1.9) and (1.14) demonstrate. However Sarkisov proposed a way to use the MMP to connect any two birational Mori fibre spaces by a sequence of four types of elementary links, see [8]. The Sarkisov program was recently shown to work in all dimensions in [13].

Note that termination of the MMP is far more subtle in dimension at least three. It is clear that we cannot keep contracting divisors. As in the case of surfaces the relative Picard number drops every time we contract a divisor and is unchanged under flips and so we can only contract a divisor finitely many times. However it is far less clear which discrete invariants improve after each flip.

Conjecture 2.8. There is no infinite sequence of flips.

The rest of this paper will be devoted to exploring (2.8).

We know that the MMP always works for toric varieties, due to the work of Reid, [35] and Kawamata, Matsuda and Matsuki, [21]. The proof is almost entirely combinatorial.

3. Local approach to termination

We review the first approach to the termination of flips. The idea is to find an invariant of X which has three properties:

(1) The invariant takes values in an ordered set I.

(2) The invariant always increases after a flip.

(3) The set I satisfies the ascending chain condition (abbreviated to ACC).

Typically the invariant is some measure of the complexity of the singularities of X. Usually it is not hard to ensure that properties (1) and (2) hold. There are many sensible ways to measure the complexity of a singularity and flips tend to improve singularities. The most subtle part seems to be checking that (3) holds as well.

The most naive invariant of any singularity is the multiplicity. If X ⊂ ℂ^{n+1} and X is defined by the analytic function f(z_{1}, z_{2}, …, z_{n}) the multiplicity m of X at the origin is the smallest positive integer such that f ∈ 𝔪^{m} = ⟨z_{1}, z_{2}, …, z_{n}⟩^{m}. If we take the reciprocal of the multiplicity then the set

I = \{ \frac{1}{m} | m ∈ ℕ \},

is naturally ordered and clearly satisfies the ACC. Unfortunately it is hard to keep track of the behaviour of the multiplicity under flips.

The idea is to pick an invariant which is more finely-tuned to the canonical divisor:

Definition 3.1. Let X be a normal quasi-projective variety. A log resolution is a projective morphism π: Y ⟶ X such that Y and the exceptional locus is log smooth, that is, Y is smooth and the exceptional locus is a divisor with simple normal crossings.

If K_{X} is ℚ-Cartier then we may write

K_{Y} + E = π^{∗}K_{X} + ∑ a_{i}E_{i},

where E = ∑ E_{i} and a_{i} are rational numbers. The log discrepancy of E_{i} with respect to K_{X} is a_{i}. The log discrepancy of X is the infimum of the a_{i}, over all exceptional divisors on all log resolutions.

We say that X is terminal, canonical, log terminal, log canonical if a > 1, a ≥ 1, a > 0 and a ≥ 0.

If V ⊂ X is a closed subset, then the log discrepancy of X at V is the infimum of the a_{i}, over all exceptional divisors whose image is V, and all log resolutions.

The log discrepancy along V is the infimum of the a_{i}, over all exceptional divisors whose image is contained in V, and all log resolutions.

Let us start with some simple examples.

Example 3.2. Let S be a smooth surface and let p ∈ S. Let π: T ⟶ S blow up p, with exceptional divisor E. Suppose we write

K_{T} + E = π^{∗}K_{S} + aE.

If we intersect both sides with E then we get

−2 = K_{ℙ^{1}} = K_{E} = (K_{T} + E) · E = (π^{∗}K_{S} + aE) · E = aE^{2} = −a.

So a = 2. It is a simple matter to check that if we blow up more over the point p then every exceptional divisor has log discrepancy greater than two. So the log discrepancy of a smooth surface is 2. It is also not hard to check that if X is not log canonical then the log discrepancy is −∞ and that if X is log canonical the log discrepancy is the minimum of the log discrepancy of the exceptional divisors of any log resolution.

If X is an affine toric variety, corresponding to the cone σ, then K_{X} is ℚ-Cartier if the primitive generators of the one dimensional faces of σ ⊂ ℝ^{n} lie in an affine hyperplane (this is always the case if σ is simplicial). In this case there is a linear functional ϕ: ℝ^{n} ⟶ ℝ which takes the value 1 on this hyperplane. The log discrepancy of any toric divisor is the value of ϕ on the primitive generator of the extremal ray corresponding to this divisor. In particular X is log terminal.

For example if X is smooth of dimension n, then X corresponds to the cone spanned by the standard generators e_{1}, e_{2}, …, e_{n} of the standard lattice ℤ^{n} ⊂ ℂ^{n}. If we insert the vector e_{1} + e_{2} then the log discrepancy of the exceptional divisor of the blow up of the corresponding codimension two coordinate subspace is 2 and this is the log discrepancy of X. If we insert the sum e_{1} +e_{2} +⋯+e_{n} this corresponds to blowing up the origin. The log discrepancy of the exceptional divisor is n and this is the log discrepancy of X at the origin.

Proposition 3.3. If π: X −−→ Y is a flip, then the log discrepancy of any divisor E never goes down and always goes up if the centre of E is contained in the indeterminacy locus of π.

Proof. See (5.11) of [21]. □

Definition 3.4 (Shokurov). Let X be a threefold with canonical singularities. The difficulty of X is the number of divisors of log discrepancy less than two.

Lemma 3.5. Let X be a threefold with canonical singularities. Then

(1) the difficulty is finite,

(2) if the difficulty is zero, then X is smooth, and

(3) the difficulty always goes down under flips.

Proof. It is easy to check that (1) holds by direct computation on a log resolution. (2) follows from the classification of canonical threefold singularities.

Let ϕ: X −−→ Y be a flip. Let C be a flipped curve, that is, a curve contained in the indeterminacy locus of ϕ^{−1}. As the log discrepancy goes up under flips, Y is terminal about a general point of C. It follows that Y is smooth along C so that there is an exceptional divisor E with centre C of log discrepancy two. The log discrepancy of E with respect to X must be less than two, by (3.3). It follows that the difficulty decreases by at least one. (3) follows easily. □

Note that (3.5) easily implies that there is no infinite sequence of flips, starting with a threefold with canonical singularities. There have been many papers which extend Shokurov’s work to higher dimensions, most especially to dimension four, for example [28], [11] and [3]. Unfortunately in higher dimensions there are infinitely many divisors of log discrepancy at most two and singular varieties of log discrepancy greater than two. It seems hard to control the situation using only the difficulty.

To remedy this situation, Shokurov has proposed some amazing conjectural properties of the log discrepancy:

Conjecture 3.6 (Shokurov). Fix a positive integer n. The set

L_{n} = { a ∈ ℚ | a is the log discrepancy of a normal variety of dimension n },

satisfies the ACC.

Conjecture 3.7 (Shokurov). Let X be a quasi-projective variety. The function

a: X ⟶ ℚ,

which sends a point x to the log discrepancy of X at x is lower semicontinuous.

Theorem 3.8 (Shokurov). Assume (3.6)_{n} and (3.7)_{n}.

Then every sequence of flips in dimension n terminates, that is, (2.8)_{n} holds.

Proof. We sketch Shokurov’s beautiful argument.

Suppose not, that is, suppose we are given an infinite sequence of flips ϕ_{i}: X_{i} −−→ X_{i+1}. Let E_{i} be the locus of indeterminacy of ϕ_{i}. Then E_{i} is a closed subset of X_{i}.

Let a_{i} be the log discrepancy of X_{i} along E_{i}. Let

α_{i} = inf{ a_{j} | j ≥ i }.

(3.3) implies that α_{i} ≤ α_{i+1}. As we are assuming (3.6)_{n} it follows that α_{i} is eventually constant. Suppose that α_{i} = a, for i sufficiently large. Then a_{i} ≥ a for all i, with equality for infinitely many i.

By assumption for each i there is a log resolution and at least one exceptional divisor F_{i} whose centre is contained in E_{i} such that the log discrepancy of F_{i} is a. Let d_{i} be the maximal dimension of the centre of any such exceptional divisor F_{i}. Pick d such that d_{i} ≤ d with equality for infinitely many i.

Let

W_{i} = { x ∈ X_{i} | x ∈ V, dim V = d, log discrepancy of X at V is at most a }.

As we are assuming (3.7)_{n}, W_{i} ⊂ X_{i} is a closed subset. Moreover some results of Ambro [4] imply that if V ⊂ W_{i} is a closed subset of dimension d then the log discrepancy of X_{i} at V is at most a with equality if V passes through the general point of W_{i}.

It is not hard to argue that as d is maximal, eventually there is an induced birational map ϕ_{i}: W_{i} −−→ W_{i+1}, which by assumption is infinitely often not an isomorphism along some subvariety of dimension d. If V ⊂ W_{i+1} is of dimension d then the log discrepancy of X_{i+1} at V is a. It follows that ϕ_{i}^{−1} must be an isomorphism along V, since the log discrepancy of X_{i} along E_{i} is a and log discrepancies only go up under flips, (3.3).

The only possibility is that ϕ_{i} must contract a subvariety of dimension d. But this cannot happen infinitely often, a contradiction. □

Note that there are more general versions of (3.6) and (3.7), which involve log pairs (X, Δ) and that Shokurov proves that if one assumes these more general conjectures then any sequence of log flips terminates. For more details see [41].

Unfortunately both (3.6) and (3.7) seem to be hard conjectures. We know (3.6)_{2} and (3.7)_{2}, by virtue of Alexeev’s classification of log canonical surface singularities. We know that

L_{3} ∩ [1, ∞) = \{ 1 + \frac{1}{r} | r ∈ ℕ ∪ \{∞\} \},

by virtue of the classification of terminal singularities due to Mori, [30] and Reid [36] and a result of Kawamata, see the appendix to [37]. Borisov [7] proved that (3.6) holds for toric varieties. Ambro proved [4] that (3.7)_{3} holds and that (3.7) holds for toric varieties.

One interesting consequence of (3.7) is the following:

Conjecture 3.9 (Shokurov). Let X be a normal quasi-projective variety of dimension n.

Then the log discrepancy of any point is at most n.

Indeed if x ∈ X, then pick a curve C which contains x and intersects the smooth locus X_{0} of X. Then x is the limit of points y ∈ C ∩ X_{0}. We have already seen that the log discrepancy of X at y is n. So if we assume (3.7)_{n} then the log discrepancy of X at x is at most n.

Note that to prove (3.9) we may assume that the log discrepancy is greater than one, that is, we may assume that X is terminal. Even though (3.9) would appear to be much weaker than (3.7), we only know that (3.9)_{3} holds by virtue of Mori’s classification of threefold terminal singularities and a result of Markushevich, [27].

4. Global approach to termination

Instead of focusing on showing that some invariant satisfies the ACC, the global approach to termination tries to use the global geometry of X. At this point it is convenient to work with:

Definition 4.1. A log pair (X, Δ) is a normal variety together with a divisor Δ ≥ 0 such that K_{X} + Δ is ℝ-Cartier.

One can define the log discrepancy and the various flavours of log terminal, just as for the canonical divisor.

Example 4.2. Let X be a toric variety and let Δ = ∑ D_{i} be the sum of the invariant divisors. Then K_{X} + Δ ∼_{ℚ} 0 so that (X, Δ) is a log pair. We may find π: Y ⟶ X a toric log resolution. Note that

K_{Y} + Γ = π^{∗}(K_{X} + Δ),

where Γ = ∑ G_{i} is the sum of the invariant divisors on Y, since both sides are zero. As π is toric, Γ contains all of the exceptional divisors with coefficient one. It follows that (X, Δ) is log canonical.

We use the following finiteness result:

Theorem 4.3 (Birkar, Cascini, Hacon, M^{c}Kernan). Let X be a smooth projective variety. Fix an ample divisor A and finitely many divisors B_{1}, B_{2}, …, B_{k} such that (X, ∑ B_{i}) is log smooth.

Then there are finitely many 1 ≤ i ≤ m rational maps ϕ_{i}: X −−→ Y_{i} such that if (b_{1}, b_{2}, …, b_{k}) ∈ [0, 1]^{k} and ϕ: X −−→ Y is a weak log canonical model of K_{X} + A + ∑ b_{i}B_{i} then ϕ = ϕ_{i} for some index 1 ≤ i ≤ m.

We have already remarked (2.7) that there are examples due to Reid of threefolds with infinitely many minimal models. The presence of the divisor A is therefore important in the statement of (4.3). However Shokurov [38] proves a similar result for threefolds, but now without the ample divisor A and shows that the same result holds in all dimensions if one knows the abundance conjecture, (5.7). In a related direction, Kawamata, [18] and Morrison [32] have conjectured that the number of minimal models is finite up to birational automorphisms of X, when X is Calabi-Yau and Δ is empty.

We use (4.3) to run a special MMP, known as the MMP with scaling. Step 0: Start with a projective variety X, an ample divisor A, a di-

visor B = ∑ b_{i}B_{i}, where (X, ∑ B_{i}) is log smooth and (b_{1}, b_{2}, …, b_{k}) ∈ [0, 1]^{k} and an ample divisor H such that K_{X} + A + B + H is nef.

Step 1: Let

λ = inf{ t ∈ [0, 1] | K_{X} + A + B + tH is nef },

be the nef threshold.

Step 2: Is λ = 0? If yes, then stop.

Step 3: If no, then there must be a curve C such that (K_{X} + A + B) · C < 0 and (K_{X} + A + B + λH) · C = 0. We can always choose C so that there is a contraction morphism π: X ⟶ Z which contracts C and there are two cases:

(i) dim Z < dim X. C is contained in a fibre. −(K_{X} + A + B) is ample on a fibre.

(ii) dim Z = dim X. In this case π is birational and there are two subcases:

(a) π contracts a divisor E.

(b) π is an isomorphism in codimension at least two.

Step 4: If we are in case (i), then stop. If we are in case (a) then replace X by Z and go back to (2). If we are in case (b) then replace X by the flip X −−→ Y and go back to (2).

Note that if H is any ample divisor then K_{X} + A + B + tH is ample for any t sufficiently large. So finding an ample divisor H such that K_{X} + A + B + H is nef is never an issue. Note also that if λ = 0 then K_{X} + A + B is nef and we have arrived at a log terminal model. The only significant difference between the MMP with scaling and the usual MMP is that we only choose to contract those curves on which K_{X} +A+B+λH is zero. With this choice, it is easy to see that we keep the condition that K_{X} + A + B + λH is nef. More to the point, every step of the MMP is a weak log canonical model of K_{X} + A + (B + λH), for some choice of λ ∈ [0, 1]. Finiteness of models, (4.3) and the fact that we never return to the same model, (3.3), implies that the MMP with scaling always terminates.

To run the MMP with scaling, we need the ample divisor A. If we start with K_{X} + Δ kawamata log terminal, we can find A ample and B ≥ 0 such that K_{X} + Δ ∼_{ℝ} K_{X} + A + B, where K_{X} + B is kawamata log terminal if and only if Δ is big. If we start with a birational map π: X ⟶ Y then every divisor is big over Y and so the MMP with scaling always applies if we work over Y.

For example, we may use the MMP with scaling to show that every complex manifold which is birational to a projective variety but which is not a projective variety must contain a rational curve. For example, one might modify Hironaka’s example, (1.15), by starting with any smooth projective threefold X with two curves intersecting transversely at two points. It is easy to find many examples which don’t contain any rational curves. But the next step involves blowing up both curves and so M contains lots of rational curves.

Shokurov [39] proved the following result assuming the full MMP and our proof is based heavily on his ideas:

Theorem 4.4 (Birkar, Cascini, Hacon, M^{c}Kernan). Let M be a smooth complex manifold. Suppose there is a proper birational map π: X ⟶ M such that X is smooth and projective.

If M does not contain a rational curve then M is projective.

Proof. Pick an ample divisor H such that K_{X} + H is ample. We run the K_{X}-MMP with scaling of H. Suppose that π: X ⟶ Y is a K_{X}- negative contraction. by a result of Miyaoka and Mori, [29], the locus contracted by π is covered by rational cuves. As M does not contain a rational curve, it follows that π is a morphism over M. In particular the (K_{X} +H)-MMP is automatically a MMP over Y. As π is birational, it follows that the MMP with scaling terminates, as observed above. At the end we have a projective variety Y such that K_{Y} is nef and a birational morphism f: Y ⟶ M. As M is smooth it follows that f is an isomorphism so that M is a projective variety. □

5. Local-global approach to termination

Even though the MMP with scaling is useful, it is becoming increasingly clear that we would still like to have the full MMP, even in the special case when Δ is big. This would be useful in the construction of the moduli space of varieties of general type. One possible approach is to try to blend both the local and the global approach to termination of flips.

We have already seen that the log discrepancy always improves under flips. However the most fundamental invariant of any singularity would seem to be the multiplicity. The log canonical threshold is a more sophisticated version of the multiplicity which takes into account higher terms and is at the same time more adapted to the canonical divisor. If X ⊂ ℂ^{n} is a hypersurface, then the log canonical threshold λ of X at the origin, is the largest t such that (ℂ^{n}, tX) is log canonical in a neighbourhood of the origin. If X has multiplicity m at the origin, then we have

\frac{1}{m} ≤ λ ≤ \frac{n}{m}.

Shokurov has conjectured that the set of log canonical thresholds should satisfy the ACC:

Conjecture 5.1 (Shokurov). Fix a positive integer n and a subset I ⊂ [0, 1] which satisfies the descending chain condition (abbreviated to DCC).

Then there is a finite set I_{0} ⊂ I such that if

(1) X is a variety of dimension n,

(2) (X, Δ) is log canonical,

(3) every component of Δ contains a non kawamata log terminal centre of (X, Δ), and

(4) the coefficients of Δ belong to I, then the coefficients of Δ belong to I_{0}.

Example 5.2. Let X ⊂ ℂ^{n} be the hypersurface given by the equation

x_{1}^{a_{1}} + x_{2}^{a_{2}} + ⋯ + x_{n}^{a_{n}} = 0.

Then the log canonical threshold is

min(\frac{1}{a_{1}} + \frac{1}{a_{2}} + ⋯ + \frac{1}{a_{n}} , 1).

It is elementary to check that these numbers satisfy the ACC.

Theorem 5.3 (Special termination; Shokurov). Assume (2.8)_{n−1}.

Let (X, Δ) be a projective log canonical pair of dimension n. Let ϕ_{i}: X_{i} −−→ X_{i+1} be a sequence of flips. Let Z_{ij} ⊂ X_{i} be the locus where the induced birational map X_{i} −−→ X_{j} is not an isomorphism. Let

Z_{i} = ⋃_{j>i} Z_{ij}.

Let V_{i} be the locus where K_{X_{i}} + Δ_{i} is not kawamata log terminal. Then V_{i} and Z_{i} eventually don’t intersect.

Note that V_{i} is a closed subset of X_{i}, whilst Z_{i} is a countable union of closed subsets of X_{i}.

Theorem 5.4 (Birkar). Assume (2.8)_{n−1} and (5.1)_{n}. Let (X, Δ) be a projective kawamata log terminal pair of dimension n.

If there is a divisor M ≥ 0 which is numerically equivalent to K_{X}+Δ, then every sequence of (K_{X} + Δ)-flips terminates.

Proof. We sketch Birkar’s ingenious argument.

Let ϕ_{i}: X_{i} −−→ X_{i+1} be a sequence of (K_{X} + Δ)-flips. Let Z_{ij} ⊂ X_{i} be the locus where the induced birational map X_{i} −−→ X_{j} is not an isomorphism. Let

Z_{i} = ⋃_{j>i} Z_{ij}.

Let Δ_{i} and M_{i} be the strict transforms of Δ and M. Note that K_{X_{i}} + Δ_{i} is numerically equivalent to M_{i}. In particular ϕ_{1}, ϕ_{2}, … is also a sequence of (K_{X} + Δ + tM)-flips for any t ≥ 0. Let

λ_{i} = sup{ t ∈ ℝ | K_{X_{i}} + Δ_{i} + tM_{i} is log canonical along Z_{i} },

be the log canonical threshold along Z_{i}. Note that λ_{i} ≤ λ_{i+1}, as log discrepancies only go up under flips. In particular if I is the set of all coefficients of Δ_{i} + λ_{i}M_{i}, then I satisfies the DCC. As we are assuming (5.1)_{n}, it follows that λ_{1}, λ_{2}, … is eventually constant. Suppose that λ_{i} = λ, for all i ≥ i_{0}. As we are assuming (2.8)_{n−1}, (5.3) implies that V_{i} ∩ Z_{i} is eventually empty, that is, the sequence of flips is finite. □

Note that we cheated a little in the proof of (5.4). Eventually K_{X_{i}} + Δ_{i} + λ_{i}M_{i} is not log canonical, so that strictly speaking (5.3) does not apply. In practice one can get around this by passing to a log terminal model. For more details, see [5].

To give a complete proof of termination of flips using (5.4), note that we need to do two things. Obviously we need to prove (5.1). However to complete the induction we need to deal with the case when K_{X} + Δ is not numerically equivalent to a divisor M ≥ 0. This part breaks up into two separate pieces.

Definition 5.5. Let X be a normal projective variety. We say that D is pseudo-effective if D is a limit of big divisors.

Conjecture 5.6. Suppose that K_{X} + Δ is kawamata log terminal.

If K_{X} +Δ is pseudo-effective then there is a divisor M ≥ 0 such that K_{X} + Δ ∼_{ℝ} M ≥ 0.

One should understand this conjecture as being part of the abundance conjecture:

Conjecture 5.7 (Abundance). Let (X, Δ) be a projective log canonical pair.

If K_{X} + Δ is nef then it is semiample.

In particular (5.6) seems very hard. One way to get around this gap in our knowledge is to assume that Δ is big. In this case we have, [6] and [42]:

Theorem 5.8 (Birkar, Cascini, Hacon, M^{c}Kernan; Siu). Suppose that K_{X} + Δ is kawamata log terminal.

If K_{X} + Δ is pseudo-effective and Δ is big then there is a divisor M ≥ 0 such that K_{X} + Δ ∼_{ℝ} M ≥ 0.

Lazić [26] and Păun [34] have since given simpler proofs of (5.8). Note that the steps of the MMP preserve the property that Δ is big. The final piece of the puzzle is to deal with the case that K_{X} +Δ is not pseudo-effective. It seems that ideas from bend and break, [29], might prove useful in this case.

Part of the appeal of this approach to termination is that (5.1) seems far more tractable than (3.6). We know (5.1) in some highly non-trivial examples. For example, Alexeev proved (5.1)_{3} [2], using boundedness of log del Pezzo surfaces whose log discrepancy is bounded away from zero. Further, de Fernex, Ein and Mustaţă, have proved the case when X is smooth, see [10] and the references therein.

We end with some speculation about a way to attack (5.1). We first note a reduction step due originally to Shokurov, see [33]. To prove (5.1)_{n} we just need to prove:

Corollary 5.9. Fix a positive integer n and a subset I ⊂ [0, 1] which satisfies DCC.

Then there is a finite set I_{0} ⊂ I such that

(1) X is a projective variety of dimension n,

(2) (X, Δ) is kawamata log terminal,

(3) Δ is big,

(4) the coefficients of Δ belong to I, and

(5) K_{X} + Δ is numerically trivial, then the coefficients of Δ belong to I_{0}.

in dimension n − 1. To this end, consider:

Conjecture 5.10. Fix a positive integer n.

Then there is a constant m such that if • X is a projective variety of dimension n,

• (X, Δ) is log canonical and log smooth,

• K_{X} + Δ is big and

• r is a positive integer such that r(K_{X} + Δ) is Cartier, then the rational map determined by the linear system |mr(K_{X} + Δ)| is birational.

Note that (5.10) closely resembles some results and conjectures stated in [12]. We note that this is slightly deceptive, since (5.10) seems quite a bit harder than these conjectures. Hopefully (5.10) has a better formulation, which is more straightforward to prove and has the same consequences. Note that if we add the condition that K_{X} + Δ is nef then the existence of m is a result due to Kollár, [23], an effective version of the base point free theorem.

The following is standard:

Lemma 5.11. Let X be a smooth projective variety of dimension n and let D be a Cartier divisor on X such that ϕ_{D} is birational.

Then ϕ_{K_{X}+(2n+1)D} is birational.

The hope is to prove (5.9)_{n} using:

Lemma 5.12. Assume (5.10)_{n}. Let I ⊂ [0, 1] be a finite set and let n be a positive integer. Suppose that I ∪ {1} are linearly independent real numbers over the rationals.

Then there is a positive real number ϵ > 0 such that if

• X is a projective variety of dimension n,

• (X, Δ) is log canonical and log smooth,

• the coefficients of Δ belong to I, and

• K_{X} + Δ is big

then K_{X} + (1 − ϵ)Δ is big.

Proof. Let m be the constant given by (5.10). By simultaneous Diophantine approximation applied to the finite set I, we may pick a positive integer r with the following properties: if a ∈ I then there is a rational number b ≥ a such that rb is an integer and

b − a < \frac{1}{2m(2n + 1)r}.

If we set

t = m(2n + 1)r,

then we may pick Θ ≥ Δ such that

∥Δ − Θ∥ < \frac{1}{2t},

where rΘ is Cartier. By (5.11),

K_{X} + m(n + 1)r(K_{X} + Θ) = (t + 1)(K_{X} + \frac{t}{t + 1}Θ),

defines a birational map. In particular

K_{X} + (1 − \frac{1}{2t}) Θ,

is big. So we may take

ϵ = \frac{1}{2m(2n + 1)r}.  □

In practice we cannot assume that the numbers I ∪ {1} are independent over the rationals and (5.12) is a simplification of our speculative argument.

References

[1] D. Abramovich, K. Karu, K. Matsuki, and J. Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531– 572 (electronic).

[2] V. Alexeev, Boundedness and K^{2} for log surfaces, International J. Math. 5 (1994), 779–810.

[3] V. Alexeev, C. Hacon, and Y. Kawamata, Termination of (many) 4dimensional log flips, Invent. Math. 168 (2007), no. 2, 433–448.

[4] F. Ambro, On minimal log discrepancies, Math. Res. Lett. 6 (1999), no. 5-6, 573–580.

[5] C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J. 136 (2007), no. 1.

[6] C. Birkar, P. Cascini, C. Hacon, and J. M^{c}Kernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468, arXiv:math.AG/0610203.

[7] A. Borisov, Minimal discrepancies of toric singularities, Manuscripta Math. 92 (1987), no. 1, 33–45.

[8] A. Corti, Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), no. 2, 223–254.

[9] , 3-fold flips after Shokurov, Flips for 3-folds and 4-folds (Alessio Corti, ed.), Oxford University Press, 2005, pp. 13–40.

[10] T. de Fernex, L. Ein, and M. Mustaţă, Shokurov’s ACC Conjecture for log canonical thresholds on smooth varieties, Duke Math. J. 152 (2010), no. 1, 93–114, arXiv:0905.3775v3.

[11] O. Fujino, Termination of 4-fold canonical flips, Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, 231–237.

[12] C. Hacon and J. M^{c}Kernan, Boundedness results in birational geometry.

[13] , The Sarkisov program, arXiv:0905.0946v1.

[14] , Extension theorems and the existence of flips, Flips for 3-folds and 4-folds (Alessio Corti, ed.), Oxford University Press, 2007, pp. 79–100.

[15] , Existence of minimal models for varieties of log general type II, J. Amer. Math. Soc. 23 (2010), no. 2, 469–490, arXiv:0808.1929.

[16] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326.

[17] K. Karu and S. Silva, On Oda’s strong factorization conjecture, arXiv:0911.4693v1.

[18] Y. Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), no. 5, 665–687.

[19] , On the extension problem of pluricanonical forms, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., vol. 241, Amer. Math. Soc., Providence, RI, 1999, pp. 193–207.

[20] , Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423.

[21] Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimal model program, Algebraic Geometry, Sendai (T. Oda, ed.), Kinokuniya-North-Holland, 1987, Adv. Stud. Pure Math, vol 10, pp. 283–360.

[22] J. Kollár, Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 113–199.

[23] , Effective base point freeness, Math. Ann. 296 (1993), no. 4, 595–605.

[24] J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), 533–703.

[25] , Birational geometry of algebraic varieties, Cambridge tracts in mathematics, vol. 134, Cambridge University Press, 1998.

[26] V. Lazić, Adjoint rings are finitely generated, arXiv:0905.2707v2.

[27] D. Markushevich, Minimal discrepancy for a terminal cDV singularity is 1, J. Math. Sci. Univ. Tokyo 3 (1996), no. 2, 445–456.

[28] K. Matsuki, Termination of flops for 4-folds, Amer. J. Math. 113 (1991), no. 5, 835–859.

[29] Y. Miyaoka and S. Mori, A numerical criterion for uniruledness, Ann. of Math. 124 (1986), 65–69.

[30] S. Mori, On 3-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66.

[31] , Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117–253.

[32] D. Morrison, Compactifications of moduli spaces inspired by mirror symmetry, Astérisque (1993), no. 218, 243–271, Journées de Géométrie Algébrique d’Orsay (Orsay, 1992).

[33] J. M^{c}Kernan and Y. Prokhorov, Threefold Thresholds, Manuscripta Math. 114 (2004), no. 3, 281–304.

[34] M. Păun, Relative critical exponents, non-vanishing and metrics with minimal singularities, arXiv:0807.3109v1.

[35] M. Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, pp. 395–418.

[36] , Canonical 3-folds, Journées de Géométrie Algébrique d’Angers (Alphen aan den Rijn) (A. Beauville, ed.), Sithoff and Noordhoff, 1980, pp. 273–310.

[37] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203.

[38] , 3-fold log models, J. Math. Sci. 81 (1996), no. 3, 2667–2699, Algebraic geometry, 4.

[39] , Letters of a bi-rationalist. I. A projectivity criterion, Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math., vol. 207, Amer. Math. Soc., Providence, RI, 1997, pp. 143–152.

[40] , Prelimiting flips, Proc. Steklov Inst. of Math. 240 (2003), 82–219.

[41] , Letters of a bi-rationalist. V. Minimal log discrepancies and termination of log flips, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 328–351.

[42] Y-T. Siu, A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring, arXiv:math.AG/0610740.

[43] , Invariance of plurigenera, Invent. Math. 134 (1998), no. 3, 661–673.

[44] J. Włodarczyk, Toroidal varieties and the weak factorization theorem, Invent. Math. 154 (2003), no. 2, 223–331.

Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112, USA

E-mail address: hacon@math.utah.edu

Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

E-mail address: mckernan@math.mit.edu