MMP for co-rank one foliation on threefolds

We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated log pairs of co-rank one on a projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated log pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated log pairs.

were shown to exist in some special cases, but not in the generality needed to run the MMP.
Our first main result is to show in greater generality that if φ R is a flipping contraction then the flip exists: Theorem 1.1 (= Theorem 6.4). Let F be a co-rank one foliation on a Q-factorial projective threefold X and let ∆ ≥ 0 such that (F , ∆) is F-dlt. Let φ : X → Y be a (K F + ∆)-flipping contraction.
Notice that F-dlt foliated pairs play the same role as dlt log pairs in the classical MMP (see Definition 3.5 for a precise definition).
Next, we turn to the question of constructing a minimal model of a foliated pair (F , ∆). As in Mori's program, existence of minimal models would follow if one could show that any sequence of flips terminates. We are unable to show termination in complete generality, but we are able to show a weaker version of termination, i.e., termination of flips with scaling, which suffices to show that minimal models exist in several cases of interest: Theorem 1.2 (= Theorem 10.3). Let F be a co-rank one foliation on a Q-factorial projective threefold X. Let ∆ = A+B be a Q-divisor such that (F , ∆) is a F-dlt pair, A ≥ 0 is an ample Q-divisor and B ≥ 0. Assume that there exists a Q-divisor D ≥ 0 such that K F + ∆ ∼ Q D.
See Section 10 for a precise definition of a minimal model. It is important to observe that Theorems 1.1 and 1.2 make no assumptions on the singularities of X other than Q-factoriality. However, as we will see by Theorem 11.3 below, the output of the MMP (and the intermediary steps of the MMP more generally) will be varieties with klt singularities. Notice also that we first prove Theorems 1.1 and 1.2 under the assumption that the foliation has non-dicritical singularities but we later prove that, if (F , ∆) is an F-dlt foliated log pair then F admits non-dicritical singularities (cf. Theorem 11.3).
Along the way to proving the termination of flips with scaling, we prove the following basepoint free theorem for foliations which we expect will be of interest. Observe that if F is a rank one surface foliation with K F nef and big then K F is in general not semi-ample. Thus, the following result is in some sense optimal: Theorem 1.3 (= Theorem 9.4). Let F be a co-rank one foliation on a Q-factorial projective threefold X. Let ∆ be a Q-divisor such that (F , ∆) is a F-dlt pair. Let A ≥ 0 and B ≥ 0 be Q-divisors such that ∆ = A + B and A is ample. Assume that K F + ∆ is nef.
Then K F + ∆ is semi-ample.

Application to F-dlt modifications and F-terminalisations.
In the study of the birational geometry of varieties, dlt modifications and terminalisations have proven to be very useful tools. The existence of these modifications follows from the MMP for varieties. We prove foliated analogues of these modifications as a consequence of our results on the foliated MMP: Theorem 1.4 (Existence of F-dlt modifications, = Theorem 8.1). Let F be a co-rank one foliation on a normal Q-factorial projective threefold X. Let (F , ∆) be a foliated pair. Then there exists a birational morphism π : Y → X and such that (G, π −1 * ∆ + ǫ(E i )E i ) is F-dlt where we sum over all π-exceptional divisors and such that where F ≥ 0.
Furthermore, we may choose (Y, G) so that (1) if Z is an lc centre of (G, Γ) then Z is contained in a codimension one lc centre of (G, Γ), (2) Y is Q-factorial and (3) Y is klt.
Theorem 1.5 (Existence of F-terminalisations, =Theorem 11.1). Let F be a co-rank 1 foliation on a Q-factorial threefold X.
Then there exists a modification π : Y → X such that (1) if G is the transformed foliation, then G is F-dlt and canonical (in particular it is terminal along sing(Y )), (2) Y is klt and Q-factorial and Using similar ideas are also able to prove the following: Theorem 1.6 (=Theorem 11.3). Let (F , ∆) be a foliated pair on a threefold X. Suppose that (F , ∆) is canonical.
Then F has non-dicritical singularities.
Observe that we do not require the smoothness of X. We expect that this result will be useful in the study of foliation singularities.
1.3. Application to foliation abundance. In [LPT11] it was proven that if X is a projective manifold with F a co-rank 1 foliation with canonical singularities and c 1 (K F ) = 0 then K F is torsion. When X is a threefold we extend this result to the log situation where we consider F together with a boundary ∆, as well as weakening the hypotheses on the singularities. Theorem 1.7 (=Theorem 12.1). Let F be a co-rank 1 foliation on a Q-factorial threefold X. Let (F , ∆) be a foliated pair with log canonical foliation singularities. Suppose that c 1 (K F + ∆) = 0.
1.4. Acknowledgements. Both the authors were funded by EPSRC.
We would like to thank J. M c Kernan, M. McQuillan, J. V. Pereira and R. Svaldi for many useful discussions. The first author would like to thank the National Center for Theoretical Sciences in Taipei and Professor J.A. Chen for their generous hospitality, where some of the work for this paper was completed.

Preliminary results
We work over the field of complex numbers C. We refer to [KM98, Section 2.3] for the classical definitions of singularities (e.g. klt) appearing in the minimal model program.
Given an effective Q-divisor ∆ on a normal variety X we write {∆} for the fractional part of ∆, i.e. ∆ − ⌊∆⌋. A Q-factorial variety is a normal variety X on which every divisor is Q-Cartier. A proper birational map f : X Y between normal varieties is a birational contraction if f −1 does not contract any divisor.
2.1. Basic definitions. A foliation on a normal variety X is a coherent subsheaf F ⊂ T X such that (1) F is saturated, i.e. T X /F is torsion free, and (2) F is closed under Lie bracket. The rank of F is its rank as a sheaf. Its co-rank is its co-rank as a subsheaf of T X .
The canonical divisor of F is a divisor K F such that O X (−K F ) ∼ = det(F ).
Alternatively, one can define a foliation by a subsheaf of the cotangent sheaf N * F ⊂ Ω 1 X which is saturated and integrable. F can be recovered as the annihilator of N * F . We call N * F the conormal sheaf of F . By abuse of notation, we denote by N * F the divisor associated to N * F , i.e. N * F = c 1 (N F ).

Foliation singularities.
Frequently in birational geometry it is useful to consider pairs (X, ∆) where X is a normal variety and ∆ is a Q-Weil divisor such that K X + ∆ is Q-Cartier. By analogy we define Definition 2.1. A foliated pair (F , ∆) is a pair of a foliation and a Q-Weil (R-Weil) divisor such that K F + ∆ is Q-Cartier (R-Cartier).
Note also that we are typically interested only in the case when ∆ ≥ 0, although it simplifies some computations to allow ∆ to have negative coefficients.
Given a birational morphism π : X → X and a foliated pair (F , ∆) on X let F be the pulled back foliation onX and π −1 * ∆ be the strict transform. We can write The rational number a(E i , F , ∆) denotes the discrepancy of (F , ∆) with respect to E i .
Notice that these notions are well defined, i.e., ǫ(E) and a(E, F , ∆) are independent of π.
Observe that in the case where F = T X , no exceptional divisor is invariant, i.e., ǫ(E) = 1, and so this definition recovers the usual definitions of (log) terminal, (log) canonical.
Definition 2.3. Given a foliated pair (F , ∆) we say that W ⊂ X is a log canonical centre (lc centre) of (F , ∆) provided (F , ∆) is log canonical at the generic point of W and there is some divisor E of discrepancy −ǫ(E) on some model of X dominating W .
Notice that in the case that ǫ(E) = 0 for all exceptional divisors over a centre the notions of log canonical and canonical coincide. In this case we will still refer to canonical centres as log canonical centres.
We also remark that any F -invariant divisor is an lc centre of (F , ∆). We have the following nice characterisation due to [McQ08, Corollary I.2.2.]: Proposition 2.4. Let 0 ∈ X be a surface germ with a terminal foliation F .
Then there exists a smooth foliation G, on a smooth surface, Y , and a cyclic quotient Y → X such that F is the quotient of G by this action.
We also make note of the following easy fact: Lemma 2.5. Let π : Y → X be a proper biratonal morphism between normal varieties. Let F be a foliation on X and let G be the foliation induced on Y . Write π * (K F + ∆) = K G + Γ.
Remark 2.6. Observe that if any component of supp(∆) is foliation invariant, then (F , ∆) is not log canonical.
We will also make use of the class of simple foliation singularities: Definition 2.7. Let F be a foliation on a smooth variety X. We say that p ∈ X is a simple singularity for F provided in formal coordinates around p, N * F is generated by a 1-form which is in one of the following two forms, for some 1 ≤ r ≤ n: (1) There are λ i ∈ C * such that and if a i λ i = 0 for some non-negative integers a i then a i = 0 for all i.
(2) There is an integer k ≤ r such that where p i are positive integers, without a common factor, ψ(s) is a series which is not a unit, and λ i ∈ C and if a i λ i = 0 for some non-negative integers a i then a i = 0 for all i. We say the integer r is the dimension-type of the singularity.
Remark 2.8. A general hyperplane section of a simple singularity is again a simple singularity. By Cano, [Can04], every foliation on a smooth threefold admits a resolution by blow ups centred in the singular locus of the foliation such that the transformed foliation has only simple singularities.
Using [AD13] it is easy to check the following: Lemma 2.9. Simple singularities (including smoothly foliated points) are canonical.
Definition 2.10. Given a foliated pair (X, F ) we say that F has nondicritical singularities if for any q ∈ X and any sequence of blow ups π : X ′ → X such that π −1 (q) is a divisor we have that each component of π −1 (q) is foliation invariant.
Remark 2.11. Observe that non-dicriticality implies that if X is smooth Definition 2.12. Given a germ 0 ∈ X with a foliation F such that 0 is a singular point for F we call a (formal) hypersurface germ 0 ∈ S a (formal) separatrix if it is invariant under F .
Note that away from the singular locus of F a separatrix is in fact a leaf. Furthermore being non-dicritical implies that there are only finitely many separatrices through a singular point. The converse of this statement is false.
Example 2.13. Let λ ∈ R. Consider the foliation F λ on C 2 generated by x∂ x + λy∂ y . For λ ∈ Q ≥0 we can see that F λ is dicritical, and otheriwse is non-dicrtical.
Even for simple foliation singularities it is possible that there are separatrices which do not converge. However, as the following definition/result of [CC92] shows there is always at least 1 convergent separatrix along a simple foliation singularity of codimension 2.
Definition 2.15. For a simple singularity of type (1), all separatrices are convergent.
For a simple singularity of type (2), around a general point of a codimension 2 component of the the singular locus we can write ω = pydx + qxdy + xψ(x p y q )λdy. The hypersurface {x = 0} is a convergent separatrix, called the strong separatrix.
Definition 2.16. Suppose X is a normal variety and F is a co-rank one foliation with non-dicritical singularities.
We say W ⊂ X (possibly contained in sing(X)) is tangent to the foliation if for any birational morphism π : X ′ → X and any (equivalently some) π-exceptional divisor E such that E dominates W we have that E is F ′ -invariant, where F ′ is the induced foliation on X ′ . We say W ⊂ X (possilby contained in sing(X)) is transverse to the foliation if for any birational morphism π : X ′ → X and any (equiva- Notice that this definition agrees with the usual definition if W is not contained in sing(X) ∪ sing(F ).

F-dlt foliated pairs and basic adjunction type results
The goal of this section is to define a new category of foliated log pair singularities, namely F-dlt pairs. These are analogous of dlt log pairs in the classical MMP and they seem to be the most suitable singularities to run a foliated MMP. In particular, we prove several properties satisfied by these pair, which we will use later on in the paper.
3.1. Foliated log smooth pairs. Definition 3.1. Given a normal variety X, a co-rank one foliation F and a foliated pair (F , ∆) we say that (F , ∆) is foliated log smooth provided the following hold: (1) (X, ∆) is log smooth.
(3) If S is the support of the non-invariant components of ∆ then for any p ∈ X, if Σ 1 , ..., Σ k are the separatrices of F at p (formal or otherwise), then S ∪ Σ 1 ∪ ... ∪ Σ k is normal crossings at p. Given a normal variety X, a co-rank one foliation F and a foliated pair (F , ∆) a foliated log resolution is a proper birational morphism π : Y → X so that exc(π) is a divisor and (G, π −1 * ∆ + i E i ) is foliated log smooth where G is the induced foliation on Y and the E i are all the π-exceptional divisors.
If X is a threefold, then such a resolution is known to exist by [Can04].
• Items (2) and (3) in Definition 3.1 imply that each component of S is everywhere transverse to the foliation, no strata of S is tangent to the foliation and no strata of sing(F ) is contained in S.
• If F is log smooth and if D is a F -invariant divisor then it is not necessarily the case that D is smooth, although it will have at worst normal crossings singularities.
Lemma 3.3. Suppose F is a co-rank one foliation on a normal variety Proof. By Lemma 2.9, since F has simple singularities, it follows that F is canonical. Now let π : Y → X be a blow up of subvariety Z ⊂ supp(∆) where Z has codimension k. Let E be the exceptional divisor. We compute the discrepancy of this blow up as follows: (1) If Z is transverse to the foliation then the discrepancy is where the inequality holds since k ≥ #{i | Z ⊂ D i } by Item (1) in Definition 3.1.
(2) If Z is tangent to the foliation but not contained in sing(F ) then the discrepancy is where the inequality holds since k ≥ #{i | Z ⊂ D i } + 1 by Item (3) in Definition 3.1. (3) If Z ⊂ sing(F ) then let m be the codimension of the minimal strata of sing(F ) containing Z. The discrepancy of the blow up is where the inequality holds since k ≥ m + #{i | Z ⊂ D i } by Item (3) in Definition 3.1. The result then follows by this computation and induction.
Note that in contrast to the classical situation, if a (F , ∆) is foliated log smooth pair then (F , ∆) may have infinitely many lc centres: Example 3.4. Let (F , D 1 + D 2 ) be a foliated log smooth pair on a threefold X, for some prime divisors D 1 and D 2 such that Z = D 1 ∩ D 2 is non-empty. Then Z is transverse to the foliation and is an lc centre of (F , D 1 + D 2 ). However, if p ∈ Z and if π : Y → X is the blow up at p with exceptional divisor E then the discrepancy with respect to E is 0 = ǫ(E) and so p is an lc centre of (F , D 1 + D 2 ). In particular, (F , D 1 + D 2 ) admits infinitely many lc centres.
Note also that if F is a foliation on a smooth projective variety X which is induced by a fibration onto a curve then any smooth vertical fibre is an lc centre.
3.2. F-dlt foliated pairs. Definition 3.5. Let X be a normal variety and let F be a co-rank one foliation on X. Suppose that K F + ∆ is Q-Cartier. We (1) each irreducible component of ∆ is transverse to F and has coefficient at most 1, and (2) there exists a foliated log resolution π : Y → X of (F , ∆) which only extracts divisors E of discrepancy > −ǫ(E).
Lemma 3.6. Let X be a Q-factorial variety and let F be a nondicritical co-rank one foliation on X. Suppose that (F , ∆) is a F-dlt pair on X and let W ⊂ X be an lc centre. Then (F , ∆) is log smooth at the generic point of W .
Proof. Suppose not. Then there exists a log resolution π : Y → X of (F , ∆) such that a(E, F , ∆) > −ǫ(E) for any π-exceptional divisor E and π is not an isomorphism at the general point of π −1 (W ). In particular, there exists a π-exceptional divisor E such that W ⊂ π(E). We may write K F Y + Γ = π * (K F + ∆) + F where Γ, F ≥ 0 are Q-divisors without any common component and F Y is the induced foliation on Y .
Assume first that W is tangent to F . Then, any π-exceptional divisor S whose centre in X is W is such that ǫ(S) = 0 and, in particular, S is contained in the support of F . Since W is an lc centre, there exists T whose centre in X is W , such that ǫ(T ) = 0 and a(T, F , ∆) = 1. Thus, the centre of T in Y is contained in the support of F . It follows Assume now that W is transverse to F . Then any π-exceptional divisor whose centre in X is W is such that ǫ(S) = 1 and, in particular, S is not contained in ⌊Γ⌋. Since W is an lc centre, there exists T whose centre in X is W , such that ǫ(T ) = 1 and a(T, F , ∆) = 0. It follows that the coefficient of Γ along a component containing the centre of T in Y is less than one and a(T, F Y , Γ) ≤ a(T, F , ∆) = 0, which contradicts Lemma 3.3.
Proposition 3.7. Let X be a Q-factorial variety and let F be a nondicritical co-rank one foliation on X. Let (F , ∆) be a F-dlt pair on X.
Then (F , ∆) has only finitely many lc centres of codimension at least two outside the support of ⌊∆⌋.
Proof. By Lemma 3.6, we have that (F , ∆) is foliated log smooth at the generic point of every lc centre. Let Z ⊂ X be a subvariety of codimension al least two which is not contained in the support of ⌊∆⌋ and such that (F , ∆) is foliated log smooth at the generic point of Z. Let π : Y → X be the blow up at Z with exceptional divisor E and suppose that Z ⊂ supp(∆). Computing as in Lemma 3.3, we see that the discrepancy of this blow up is > −ǫ(E). Computing inductively we see that every divisor dominating Z has discrepancy > −ǫ(E).
Thus, every lc centre of (F , ∆) must also be an lc centre of (F , 0). Keeping in mind that F has simple singularities at the general point of Z, a straightforward computation shows that the lc centres of (F , 0) are strata of sing(F ).
We make note of the following easy observation: Lemma 3.8. Let X be a normal variety and let F be a co-rank one foliation on X. Suppose that (F , ∆) is a F-dlt pair on X and that X X ′ is a sequence of steps of a (K F + ∆)-MMP. Let (F ′ , ∆ ′ ) be the induced foliated pair on X ′ .
Lemma 3.9. Suppose that X is a Q-factorial threefold and let F be a non-dicritical co-rank one foliation on X. Let C ⊂ X be a curve tangent to F and suppose that (F , ∆) is F-dlt. Then (1) (F , ∆) is canonical along C.
(2) If in addition C ⊂ sing(X) then (F , ∆) is terminal along C.
Proof. Item (1) follows from the observation that every divisor E dominating C on some log resolution must be foliation invariant. If (F , ∆) is not terminal along C then C is an lc centre. Thus, item (2) follows from Lemma 3.6.
Corollary 3.10. Let X be a Q-factorial threefold and let F be a corank one foliation on X. Suppose that (F , ∆) is F-dlt, let S be an invariant divisor and let n : S n → S be the normalisation. Let T be the sum of every invariant divisor meeting S, formal or otherwise. Write (K F + ∆)| S n = K S n + Θ and (K X + ∆ + S + T )| S n = K S n + Θ ′ .
Then Θ ′ ≤ Θ with equality along centres contained in sing(X). Moreover, for any curve C in S along which S is a strong separatrix, the coefficients of Θ and Θ ′ coincide.
Proof. By Lemma 3.9 we see that if C ⊂ sing(X)∩S then F is terminal along C. In this case, the result follows as in the proof of [Spi17, Lemma 8.6].
Lemma 3.11. Let X be a Q-factorial threefold and let F be a co-rank one foliation on X. Suppose that (F , ∆) is F-dlt and let C ⊂ X be a curve tangent to F .
Then, there exists a germ of a separatrix S containing C. Moreover, either (1) F is terminal at the generic point of C or (2) X is smooth at the generic point of C and there are 2 (formal) separatrices along C.
Proof. Lemma 3.9 implies that F is canonical along C. Thus, [Spi17, Corollary 5.4, Corollary 5.5 and Corollary 5.6] implies that there exists a germ of a separatrix S containing C. If C ⊂ sing(X) then F is terminal along C by Lemma 3.9. So assume that X is smooth at the generic point of C. If F is not terminal at the generic point of C, then C is an lc centre and Lemma 3.6 implies that (F , ∆) is log smooth at the general point of C. This implies then that F has simple singularities at the generic point of C, and hence has two (possibly formal) separatrices.
(1) In general, a canonical non-terminal singularity may only admit a single separatrix formal or otherwise. Thus, canonical does not imply F-dlt.
(2) However, log terminal does imply F-dlt (keeping in mind that in general canonical does not imply log terminal for foliations). Lemma 3.13. Let X be a normal variety and let F be a co-rank one foliation on X. Let ∆ = A + B be a Q-divisor such that (F , ∆) is a F-dlt pair, A ≥ 0 is an ample Q-divisor and B ≥ 0.
Then there exist Q-divisors A ′ , B ′ ≥ 0 such that A ′ is ample and if Proof. Let δ > 0 be sufficiently small, so that A + δB is ample. Let B ′ = (1 − δ)B and let 0 ≤ A ′ ∼ Q A + δB be sufficiently general such that, if ∆ ′ = A ′ + B ′ then (F , ∆ ′ ) is F-dlt, ⌊∆ ′ ⌋ = 0 and the support of A ′ does not contain the centre of any divisor E such that a(E, F , ∆) = −ǫ(E). Thus, the claim follows easily.
Lemma 3.14. Let X be a Q-factorial variety and let F be a co-rank one foliation on X. Let ∆ = A + B be a Q-divisor such that (F , ∆) is a F-dlt pair, A ≥ 0 is an ample Q-divisor and B ≥ 0. Let ϕ : X X ′ be a sequence of steps of the (K F + ∆)-MMP and let F ′ be the induced foliation on X ′ .
Proof. By Lemma 3.13, we may assume that ⌊∆⌋ = 0. We may also assume that ϕ : X X ′ is a (K F + ∆)-flip (resp. (K F + ∆)-divisorial contraction). Let H ≥ 0 be a general ample Q-divisor on X ′ . After possibly replacing H by a smaller multiple, we may assume that if H X is the strict transform of H in X then A − H X is ample. In particular, by Proposition 3.7, there exists an effective Q-divisor C ∼ Q A − H X and ǫ > 0 sufficiently small such that (F , ∆ + ǫC) is F-dlt and ϕ is still a (K F + ∆ + ǫC)-flip (resp. (K F + ∆ + ǫC)-divisorial contraction). Thus, if F ′ is the foliation induced on X ′ , then Lemma 3.8 implies that (F ′ , ϕ * (∆ + ǫC)) is F -dlt. In particular ϕ * C does not contain any lc centre of (F ′ , ϕ * ∆).
The second claim is a direct consequence of Lemma 3.6.
The final claim follows by noting that if each D i is smooth in codimension 1 then a log resolution of (F , ∆) is also a log resolution of (X, ∆ + D).
In fact, Lemma 3.15 remains true even if X is just a formal germ about a point.
From this we can deduce an adjunction statement: Lemma 3.16 (Adjunction). Let X be a Q-factorial threefold, let F be a co-rank one foliation with non-dicritical singularities. Suppose that (F , ǫ(S)S + ∆) is lc (resp. lt, resp. F-dlt) for a prime divisor S and a Q-divisor ∆ ≥ 0 on X which does not contain S in its support. Let ν : S ν → S be the normalisation and let G be the restricted foliation to S ν .
Proof. The first claim follows from [Spi17, Proposition 3.3 and the proof of Lemma 5.9]. We now prove the second claim. Let π : Y → X be a foliated log resolution of (F , ǫ(S)S + ∆), let F Y be the foliation induced on Y and let T be the strict transform of S in Y . If (F , ǫ(S)S + ∆) is F-dlt, then we choose π so that a(E, F + ǫ(S)S, ∆) > −ǫ(E) for any π-exceptional divisor E.
Suppose that ǫ(S) = 1 and note that, by (1) of Remark 3.2, we have that T is everywhere transverse to F Y . Thus, we maywrite where ∆ ′ is the strict transform of ∆ in Y , a i ∈ Q and the sum is taken over all the π-exceptional divisors. By [Spi17, Corollary 3.2], restricting to T gives where φ : T → S is the induced morphism and G T is the restricted foliation to T . By assumption a i ≤ ǫ(E i ) (resp. a i < ǫ(E i )). To prove our result we need to show that a i ≤ ǫ(E i | T ) (resp. a i < ǫ(E i | T )).
Suppose for sake of contradiction that for some i, we have ǫ(E i ) = 1, ǫ(E i | T ) = 0 and a i > 0 (resp. a i ≥ 0). In this case, consider the blow up of Y at E i ∩ T and let F be the exceptional divisor. Notice that F is invariant and so ǫ(F ) = 0. However, this is a blow up of discrepancy ≤ −a i < ǫ(F ) (resp. ≤ −a i ≤ ǫ(F ), resp. ≤ −a i < ǫ(F )), and so we see that (F Y , T + ∆ ′ + a i E i ) is not lc (resp. lt, resp. F-dlt), hence (F , S + ∆) is not lc (resp. lt, resp. F-dlt).
To prove the second claim, we may work in a formal neighborhood of S. Let T be the sum of all the (formal) separatrices around S. Observe that if sing(F ) ∩ S is normal then each component of T is smooth in codimension 1. Then by Lemma 3.15 we know that (X, S + T + ∆) is lc (resp. lt, resp. dlt), furthermore, as in the proof of [Spi17, Lemma 8.6], it follows that the different with respect to (X, S + T + ∆) is exactly Θ ′ . We then apply usual adjunction.

F-dlt modification.
Definition 3.17. Let F be a co-rank one foliation on a normal projective threefold X. Let (F , ∆) be a foliated pair. A F-dlt modification for the foliated pair (F , ∆) is a birational morphism π : is F-dlt where we sum over all π-exceptional divisors and In particular, if (F , ∆) is lc then π only extracts divisors E i of discrepancy −ǫ(E i ).
Theorem 8.1 below will imply the existence of a F-dlt modification for any foliated pair (F , ∆).
3.5. F-dlt cone and contraction theorem. In [Spi17] the cone theorem is proved under the hypothesis that X is Q-factorial, (F , ∆) has canonical singularities and the contraction theorem is proved under the additional hypothesis that (F , ∆) is terminal along sing(X).
In fact, it is possible to prove the cone and contraction theorem under the hypotheses that (F , ∆) is F-dlt. Even better, it is possible to prove the cone theorem in the case that X is not necessarily Q-factorial but (X, D) is klt for some D ≥ 0. We explain the required modifications to the cone theorem first: Since (X, D) is klt there exists a small Q-factorialization π : Y → X. Write K G + Γ = π * (K F + ∆), where G is the induced foliation on Y . Observe that (G, Γ) is F-dlt. Notice that (K G + Γ) · C = 0 for every π-exceptional curve C so if R is a (K F + ∆)-negative extremal ray then there exists a (K G + Γ)-negative extremal ray R ′ such that π * R ′ = R. Thus, by replacing (F , ∆) by (G, Γ) we may freely assume that X is Q-factorial.
As in the proof of [Spi17,Lemma 4.4], it follows that if (F , ∆) is log canonical then every (K F + ∆)-negative extremal ray is spanned by a curve C.
We recall the following result from [Spi17].
Lemma 3.18. Let X be a threefold and let F be a non-dicritical corank 1 foliation on X.
Proof. This is proven in [Spi17,Lemma 8.7] under the assumption that X is Q-factorial and klt. However, one can observe that the proof does not rely on either of these hypotheses.
By Lemma 3.9, if C is tangent to F then (F , ∆) is canonical at the generic point of C and we have reduced to the cone theorem in [Spi17, Theorem 7.1].
Thus, we have: Theorem 3.19. Let X be a normal projective threefold and let F be a co-rank one foliation with non-dicritical singularities. Suppose that (X, D) is klt for some D ≥ 0. Let (F , ∆) be a F-dlt pair and let H be an ample Q-divisor.
Then there exist countable many curves ξ 1 , ξ 2 , . . . such that Furthermore, for each i, either ξ i is a rational curve tangent to F such that (K F + ∆) · ξ ≥ −6. and if C ⊂ X is a curve such that In particular, there exist only finitely many (K F + ∆ + H)-negative extremal rays. Now suppose that X is Q-factorial. To explain the modifications to the contraction theorem we recall the following definition: Definition 3.20. Given an extremal ray R ⊂ NE(X) we define loc(R) to be the set of all those points x such that there exists a curve C with x ∈ C and [C] ∈ R.
If loc(R) = X, then, as in [Spi17, Theorem 8.9], it follows that R is in fact K X -negative and so the contraction exists.
If loc(R) = D a divisor and D is transverse to the foliation, then as in [Spi17,Lemma 8.11], it follows that R is (K X + ∆)-negative. If D is invariant then Corollary 3.10 implies that R is (K X + ∆ + D)-negative. In either case, the contraction exists by Lemma 3.15 and the existence of (K X + D)-negative divisorial contractions.
If loc(R) = C a curve, then the contraction that is proven to exist in [Spi17, Lemma 8.16] in the category of algebraic spaces only requires that (F , ∆) be log canonical and so works for F-dlt pairs.
Remark 3.21. We will return to the problem of constructing contractions in the non-Q-factorial case in Section 8.1.

Approximating formal divisors
One of the main difficulties to prove the existence of flips for foliated log pars (F , ∆), as in Theorem 6.4, is due to the fact that in the singular settings, some of the separatrices through a curve C which is tangent to F , are defined only in a formal neighbourhood of the curve C. To this end, since the MMP for formal schemes is still unknown, we study some application of Artin and Elkik's approximation theorems.
We begin by recalling the following definition: Given a pair (A, J) of a ring A and an ideal J contained in the Jacobson radical of A, it is possible to define the henselisation of (A, J) as in [Ray70,Chapitre VIII]. Note that we do not require that A is a local ring or that J is a prime ideal.
The next result is [Elk73, Theorem 3 (see also the paragraph below Theorem 3)]: Then there exists an A-module M such that M ⊗ A A is isomorphic to M.
Furthermore, for any positive integer k, we may choose M so that if such that L = L mod J k and such that the isomorphism between M and M ⊗ A is induced by automorphisms of A p and A q congruent to the identity modulo J k .
be a section whose associated divisor is V . We may assume that there is a presentation such that s is the image of (1, 0, ..., 0) in M . Approximating this presentation by we define s to be the image of (1, 0, ..., 0) in M. In particular, we see that s = s mod J k considered as sections of M . Let V be the divisor associated to s and so M = O(V ) and V = V mod J k . Now, suppose that m > 1 is the Cartier index of O(V ) away from W . By Lemma 4.5 below, we may find a possibly ramified cover σ : Spec(B) → Spec(A) such that the ramification locus is some very general ample divisor and such that (σ * O(V )) * * is a line bundle away from σ −1 (W ) (note that in general, it may not be possible to find a cover which isétale in codimension one). Perhaps passing to a higher cover we may assume that σ is Galois with Galois group G.
Note that J ′ = B ⊗J is the ideal corresponding to σ −1 (W ). Let B be the J ′ -adic completion of B. Observe that (B, J ′ ) is a henselian couple. We may find a reflexive sheaf M ′ on Spec(B) such that M ′ ⊗ B ∼ = (σ * O(V )) * * . Let s be as above and let t = σ * s.
As before, for any positive integer ℓ, we can approximate t by a section t of M ′ such that t = t mod J ′ℓ . Perhaps replacing t by 1 #G g∈G g · t we may assume that t is also G-invariant. Let V ′ be the divisor associated to t and so Let L be a reflexive sheaf on Spec(A) such that L ⊗ A = O(mV ). Then we have (σ * L) * * ∼ = O(mV ′ ). Thus (σ * L) * * has a G-invariant section t ⊗m which descends to a section η of L.
Lemma 4.5. Let (A, J) be a henselian couple and A the J-adic comple- Then there exists a finite morphism σ : Proof. Let X = Spec(A) and let m be the Cartier index of O(V ) away from W . Let L be a reflexive sheaf on X such that L ⊗ A ∼ = O(mV ) and whose existence is guaranteed by Theorem 4.3. Let . × X V n and let K(V ) be the field of functions of V and notice that K(V ) is a finite extension of K(A). Letting B be the integral closure of A in K(V ) and letting σ : Spec(B) → Spec(A) be the natural map gives a cover with the desired properties.
Definition 4.6. Let (A, J) be a henselian couple, A the J-adic completion and let ∆ ′ be a Q-divisor on Spec( A). We say that a Q-divisor There exists an n 0 such that if n ≥ n 0 and if ∆ is an approximation of Θ mod J n then (Spec( A), ∆) is klt (resp. lc).
Proof. Let π : Y → Spec( A) be a log resolution of (Spec( A), Θ) and let E = π −1 (V (J)). Perhaps passing to a higher resolution we may assume that E is a divisor.
Let D be a component of Θ. We may write where D ′ is the strict transform of D and E i are π-exceptional and a i ≥ 0. Choose r larger than a i for all i and for all the components D of Θ. Next, pick n 0 so that This choice of r, n 0 guarantees that if D is a component of supp(Θ) and D is an approximation of D mod J n where n ≥ n 0 that if we write Thus, if ∆ is an approximation of Θ mod J n for n ≥ n 0 then π is also a log resolution of (Spec( A), ∆). Furthermore, if we let ∆ ′ (respectively Θ ′ ) be the strict transform of ∆ (respectively Θ) we see that ∆ ′ | E = Θ ′ | E which implies that the two are π-numerically equivalent and hence the discrepancies of (Spec( A), Θ) and (Spec( A), ∆) are the same and the result follows.

Approximating formal separatrices
In this section, we work in the following set up: Let X be a Q-factorial and klt threefold. Let F be a co-rank one foliation on X with non-dicritical singularities and let (F , ∆) be a F-dlt foliated pair. Let f : X → Z be a birational contraction and p ∈ f (exc(f )) be a closed point. We assume that D : Let f : X → Z be the completion of f along the fibre f −1 (p), and let F be the formal foliation.
Lemma 5.1. Let X be a normal threefold. Let F be a co-rank one foliation on X with non-dicritical singularities. Let D ⊂ X be a subvariety tangent to F , let q ∈ D be a general point and let S q be a separatrix at q (possibly formal).
Then there exists a F-invariant formal subscheme S on X which agrees with S q near q.
Proof. Let π : W → X be a high enough foliated log resolution so that π −1 (D) = E is a divisor. By non-dicriticality we see that E is foliation invariant. Letπ : W → X be the completion of W and X along E and D respectively.
There exists a point q ′ ∈ sing(F W ) corresponding to the formal separatrix S q , and we may find a formal separatrix S ′ q ′ at q ′ which is the (formal) strict transform of S q . The arguments in [CC92,§IV] (in particular the proof of [CC92, Theorem IV.2.1]) and in [Spi17, Corollary 5.4, Corollary 5.5 and Corollary 5.6] show that we can extend S ′ q ′ to a formal subscheme S ′ of W .
Let I S ′ ⊂ O W be the ideal sheaf corresponding to S ′ . By the proper mapping theorem for formal schemes, [Gro63,3.4.2],π * I S ′ is a coherent sheaf, and sinceπ * O W = O X we see that it is in fact an ideal sheaf corresponding to a formal subscheme S ⊂ X.
Since being an invariant divisor can be checked locally, it suffices to check S is a formal invariant divisor in the case where X is affine.
If X is affine, then let X = Spec(O X ) and let W = W × X X and let π : W → X be induced map. By the Grothendieck existence theorem, S ′ corresponds to a closed subscheme of W denoted S ′ andŜ correspond to a closed subscheme of X denoted S. The above construction gives usπ * S ′ = S and so we see that S is a divisor on X and is F -invariant. The theorem on formal functions tells us that the completion of S along D is exactly S, and our result follows.
Remark 5.2. Observe that by Lemma 3.11, if D i is component of D then at a general point q ∈ D i there is a separatrix at q containing D i . Lemma 5.3. Let S ⊂ X be any formal divisor. Fix an integer n > 0.
Then there exists anétale morphism σ : Z ′ → Z and a divisor S ′ on We may assume that Z = Spec(B) is affine. Let (A, m) be the henselisation of B at p, and let A be the formal completion of B at p. Let X = X × Spec(B) Spec( A). By the Grothendieck existence theorem there exists a divisor S on X such that S| X = S.
Let f : X → Spec( A) be the induced morphism. By the proper mapping theorem, we have that V := f * S is a divisor on Spec( A).
Let D = f −1 (p). Pick a positive integer k large enough so that where n ′ is a sufficiently large positive integer so that where we run overétale morphisms (Spec(B ′ ), q) → (Spec(B), p) sending q to p. Thus, we see that there exists someétale cover Spec(B ′ ) → Spec(B) and a divisor V ′ on Spec(B ′ ) which agrees with V when pulled back to Spec(A). Let S ′ be the strict transform of V ′ on X ′ . Then we have that S ′ =τ * S mod I n D ′ , as required.
Lemma 5.4. Notation as above. Let S 1 , ..., S k be any collection of The result follows by combining Lemma 4.7 and Lemma 3.15.

Constructing the flip
Through out this section, we assume that X is a klt threefold, F is a co-rank one foliation on X with non-dicritical singularities and (F , ∆) is F-dlt foliated pair. The goal of this section is to show that if f : X → Z is a flipping contraction induced by a (K F + ∆)-negative extremal ray R, then the (K F + ∆)-flip exists. The basic idea is to reduce the (K F + ∆)-flip to a (K X + ∆)-flip for some klt pair (X, ∆).
Recall that in [Spi17] it was proven that the flipping contraction exists in the category of algebraic spaces, but it was not shown there that Z is projective or that ρ(X/Z) = 1. We will show that both of these hold here.
6.1. Set-up. Let X be a Q-factorial projective threefold and let F be a co-rank one foliation on X with non-dicritical singularities. Suppose that (F , ∆) is a F-dlt foliated pair. Let f : X → Z be a (K F + ∆)negative flipping contraction. By Lemma 3.18 , it follows that exc(f ) is tangent to the foliation.
Then the D-flip exists.
Proof. The existence of the flip is equivalent to the O Y -algebra being finitely generated. Finite generation of an algebra can be checked etale locally and the result follows.
By Lemma 6.1, we may assume below that ⌊∆⌋ = 0. Let S k be the collection of all the separatrices on X containing some of the curves contracted by f , formal or otherwise, and whose existence is guaranteed by Lemma 3.11 and Lemma 5.1 (see also Remark 5.2). By Lemma 5.3, we may find a diagram where σ : Z ′ → Z isétale and divisors S ′ k on X ′ which approximate the S k to some arbitrarily high (but fixed) order. Let F ′ be the foliation induced on X ′ .
Note that X ′ is klt. Let g : Y → X ′ be a small Q-factorialization of Let U be a small analytic neighborhood around C. We can find Q- In fact, we can can make arbitrarily many sufficiently general choices for the D i .
Let S k be the strict transform of S ′ k on Y . Notice that by Corollary 3.10 and by [Spi17, Corollary 5.5], we have that for all i Thus, for an appropriate choice of rational numbers a i ≥ 0, we may assume that is klt for 0 < ǫ ≤ 1. So by choosing the D i generally enough and the a i small enough we may Since X → Z contracts only a single extremal ray and −(K F + ∆) is relatively ample, there exists λ ∈ Q such that λ(K F +∆) ≡ f (K X +∆). Since g is small and τ and τ ′ areétale, it follows that λ( 6.2. Existence of the flip. Below, we use the same notation as in the previous subsection. Lemma 6.3. Set up as above. Then ρ(X/Z) = 1 and, in particular, Z is projective.
Proof. Let D be a Q-Cartier divisor on X which is f -numerically trivial.
In order to show that ρ(X/Z) = 1, it is enough to show that there exists a Q-Cartier divisor M on Z such that f * M = D. Indeed, since f : X → Z is a flipping contraction induced by an extremal ray R, the claim implies that the sequence Since the descent problem above isétale local, it suffices to show that D ′ = (h ′ ) * M ′ for some Q-Cartier divisor on Z ′ . The existence of M ′ follows by applying the classical relative base point free theorem to the pair (Y ′ , is h ′ -big and nef for small ǫ, we have that D ′ is h ′ -semi-ample. Thus, by definition, there is some n ≫ 0 and a Cartier divisor L on Z ′ such that (h ′ ) * L = nD ′ . Thus, we may choose M ′ = 1 n L and the claim follows. Assume now that D = H R is a nef Q-Cartier divisor on X which defines a supporting hyperplane for R in NE(X) and let M be the induced Q-Cartier divisor on Z. Projectivity then follows by noting that, for any subvariety V of Z we have M dim V · V > 0. Indeed, M is ample by the Nakai-Moishezon criterion and so Z is projective.
Theorem 6.4. Let X be a Q-factorial projective threefold and let F be a co-rank one foliation on X with non-dicritical singularities. Suppose that (F , ∆) is a F-dlt foliated pair. Let f : X → Z be a (K F + ∆)negative flipping contraction.
Proof. By Lemma 6.2 it suffices to construct the flipétale locally on the base. Thus, working over Z ′′ , we see that to construct the flip it suffices to produce an ample model for However, we know that for ǫ > 0 sufficiently small Since h ′ : Y ′ → Z ′′ is small, we know that c is small and so (Y ′+ , c * (∆ ′ + (1−ǫ)A)) is in fact klt. Projectivity and Q-factoriality follow easily.
Remark 6.5. If one is so inclined this can all be done in the analytic topology around the flipping curves. The relevant classical log MMP is known to exist by [Nak87].
6.3. (K F + ∆)-negative contractions are extremal. We recall the following definition: Definition 6.6. A proper birational moprhism φ : X → Y between normal projective varieties is said to be extremal if (1) X is Q-factorial and (2) if D, E are Cartier divisors on X then there exists a, b ∈ Z such that aD ∼ φ bE.
Theorem 6.7. Let X be a normal Q-factorial projective threefold and let F be a co-rank one foliation on X with non-dicritical singularities. Let (F , ∆) be a F-dlt pair and let R be a (K F + ∆)-negative extremal ray.
Then the contraction associated to R φ R : X → Y exists in the category of projective varieties and ρ(X/Y ) = 1. In particular, if φ R is birational, then it is extremal.
Proof. If φ R is a fibre type contraction or a divisorial contraction then the theorem is known by [Spi17, Theorem 8.9 and Theorem 8.12].
If φ R is a flipping contraction then the result follows by Lemma 6.3.

Special termination
The goal of this section is to show the following: Theorem 7.1 (Special Termination). Let X be a Q-factorial quasiprojective threefold. Let (F , ∆) be an F-dlt pair and suppose F has non-dicritical singularities. Let be an infinite sequence of (K F + ∆)-flips and let (F i , ∆ i ) be the induced foliated pair on X i . Then after finitely many flips, the flipping and flipped locus are disjoint from any lc centres of (F i , ∆ i ).
Note that the result and the some of the proofs below were inspired by Shokurov's special termination in the classical setting [Sho03] (see also [Cor05,§4.2]).
We begin with the following: Lemma 7.2. Let X be a Q-factorial quasi-projective threefold. Let (F , ∆) be an F-dlt pair and suppose F has non-dicritical singularities.
Suppose that there exist infinitely many F -invariant divisors. Then any sequence of (K F + ∆)-flips terminates.
Proof. Since the intersection of two invariant divisors is contained in sing(F ) and since F has non-dicritical singularities, it follows that there exist infinitely many pairwise disjoint F -invariant divisors. By [Per06, Theorem 2], there exists a morphism f : X → C onto a curve C such that F is induced by f and, in particular, where the sum is taken over all the vertical irreducible divisors and ℓ D denotes the multiplicity of the fibre f −1 (f (D)) along D. Thus where Γ is the sum of all the vertical prime divisors which are contained in a non-reduced fibre. Since (F , ∆) is F-dlt and since any component of Γ is F -invariant, Lemma 3.15 implies that (X, ∆+Γ) is log canonical. Note that if X X ′ is a (K F + ∆)-flip and the flipping curve ξ is horizontal, then F · ξ > 0 for any general fibre F of f and, in particular, the strict transform F ′ of F on X ′ contains the flipped curve ξ ′ , contradicting the fact that the induced foliation F ′ on X ′ has non-dicritical singularities. Thus, we may assume that any sequence of (K F + ∆)-flips preserves the fibration onto the curve C. In particular, any sequence of (K F + ∆)-flips is also a sequence of (K X + ∆ + Γ)-flips. Thus, termination follows from termination of three-dimensional log canonical flips.
Thus, from now on, we assume that F admits at most finitely many invariant divisors. The proof proceeds in two steps. We first consider the case of lc centres transverse to the foliation. We then handle the case of lc centres tangent to the foliation by induction on dimension: supposing the statement is true for all d − 1 dimensional lc centres, we then prove it for all d dimensional lc centres.
We will need the following consequence of the negativity lemma: Lemma 7.3. Let φ : X X ′ be a birational map between normal varieties and let where Y is a normal variety and f and f ′ are proper birational morphisms. Let (F , ∆) be a foliated pair on X and let (F ′ , ∆ ′ ) be the induced foliation pair on X ′ . Assume that Then, for any valuation E on X, we have Moreover, the inequality holds if f or f ′ is not an isomorphism above the generic point of the centre of E in Y .
Proof. The proof is the same as [KM98, Lemma 3.38].
7.1. Log canonical centres transverse to the foliation.
Proposition 7.4. Let X be a Q-factorial quasi-projective threefold. Suppose that (F , ∆) is F-dlt and let W ⊂ X be an lc centre transverse to F . Let φ : X X + be a flip and let ∆ + = φ * ∆. Suppose that W is not contained in the flipping locus Z. Let W + be the strict transform of W and Z + be the flipped locus.
Then W + ∩ Z + ⊂ W + is not a divisor.
Proof. Suppose for sake of contradiction that W + ∩ Z + =: D ⊂ W + is a divisor. Let F + be the foliation induced on X + and let G + be the foliation restricted to W +,ν where W +,ν → W + is the normalisation. By Lemma 3.6 and Lemma 3.3, it follows that W + is contained in the support of ⌊∆⌋. Thus, by Lemma 3.16, we may write where (G + , Θ + ) is log canonical. By Lemma 7.3, applied to the map W ν W +,ν , it follows that the coefficient of D in Θ + is strictly positive, however Lemma 3.18 implies that D is G + -invariant, and this is a contradiction of the fact that (G + , Θ + ) is log canonical.
Then, after finitely many flips, the flipping locus is disjoint from all the lc centres transverse to the foliation.
Proof. Observe that by Lemma 7.3 and the fact that there are only finitely many lc centres transverse to the foliation, we may assume that no lc centre transverse to the foliation is contained in the flipping locus. Let φ : X X + be a flip. Suppose that the flipping locus meets some lc centre of (F , ∆) transverse to the foliation. Then since (F , ∆) is F-dlt, it meets some divisorial component W of ⌊∆⌋. Thus, to prove our result it suffices to show that, for any component W of ⌊∆⌋, after finitely many flips the flipping locus is disjoint from W .
So suppose that W meets the flipping locus and let ψ : W W + be the induced map. Let G denote the restricted foliation and write where, by Lemma 3.16, (G, Θ) is F-dlt. In particlar, W is klt. By Proposition 7.4, none of the curves in the flipped locus is contained in W + . Thus, ψ is a birational contraction. If ψ does not contract any divisors, then there exists a curve Z contained in the flipping locus, such that Z ∩ W = ∅ but Z is not contained in W . Then Z · W > 0 and so if Z + is a flipped curve we must have Z + · W + < 0 implying that Z + ⊂ W + , a contradiction.
Thus, ψ contracts a divisor at each flip. In particular, each flip reduces ρ(W ) by 1 and we can only have finitely many such flips. By Corollary 7.5, it suffices to show that the flipping locus is eventually disjoint from lc centres which are tangent to the foliation.
Lemma 7.6. After a finite sequence of flips, if Z if an lc centre and C is a flipping curve then Z is not contained in C.
Proof. By Corollary 7.5, after finitely many flips we may assume that the flipping locus is disjoint from all lc centres transverse to the foliation, in particular, it is disjoint from ⌊∆⌋.
By Proposition 3.7, there are only finitely many lc centres of (F , ∆) not contained in ⌊∆⌋ and so the claim follows from Lemma 7.3.
As we mentioned earlier, given a F-dlt pair (F , ∆) on a Q-factorial quasi projective threfold X, we denote by d the minimal dimension of an lc centre of (F , ∆) which is tangent to F and intersects the flipping locus of a (K F + ∆)-flip. Our goal is to show that there can be only finitely many flips with d = 0, 1 or 2.
7.2. Special termination in dimension d = 0. This follows directly from Lemma 7.6 7.3. Special termination in dimension d = 1.
Lemma 7.9. Let X be a Q-factorial quasi-projective threefold. Suppose that (F , ∆) is F-dlt. Let C be a 1-dimensional lc centre of (F , ∆) tangent to F . Write Then Θ ≥ 0 and ⌊Θ⌋ is supported on the lc locus of (F , ∆).
Proof. By Lemma 3.6, it follows that (F , ∆) is log smooth at the general point of C. By Lemma 3.11, there exists a germ of a separatrix S containing C and if C ⊂ sing(F ), then we choose S to be the strong separatrix. Then it is easy to check that for some Q-divisor ∆ S ≥ 0, whose support does not contain C.
Note that (K S + C + ∆ S )| C ν = K C ν + Θ. Thus, by usual adjunction, the non-klt locus of (C ν , Θ) is supported on the non-kltt of (S, C +∆ S ). We claim these centres are contained in lc centres of (F , ∆). Let P ∈ S, by Lemma 3.16 we see that if (F , ∆) is log terminal at P then (S, C + ∆ S ) must also be log terminal at P , and so the non-klt locus of (S, C + ∆ S ) must be contained in lc centres of (F , ∆).
Corollary 7.10. Set up as above. Then after finitely many flips the flipping locus is disjoint from all 1-dimensional lc centres.
Proof. Let I be the set of coefficients of Θ. Using the same notation as in Lemma 7.9, it follows by Lemma 7.8 and Lemma 3.18 that the coefficients of {Θ} take values in D(I) and that after a flip Θ strictly decreases. However, by Lemma 7.9 and since we are assuming that there are no zero dimensional lc centre intersecting the flipping locus, it follows that the flip is an isomorphism near ⌊Θ⌋ and the result follows.

Special termination in dimension d = 2.
Lemma 7.11. Set up as above. Then after finitely many flips the flipping locus is disjoint from all 2-dimensional lc centres.
Proof. Let I ⊂ [0, 1] be a finite set containing the coefficients of ∆. Let S be a two dimensional lc centre intersecting the flipping locus. By Corollary 7.5, we may assume that S is F -invariant and, by Lemma 3.16, we may write (K F + ∆)| W = K W + Θ for some Q-divisor Θ ≥ 0 where W → S is the normalisation and such that (W, Θ ′ := ⌊Θ⌋ red + {Θ}) is lc and (W, (1 − ǫ)Θ ′ ) is klt for 0 < ǫ < 1. Note that the coefficients of Θ ′ belong to D(I).
Suppose first that ψ −1 contracts a divisor D ⊂ S + . Let Z ⊂ S be the centre of D on S. By induction we know that Z is not contained in ⌊Θ ′ ⌋. It follows that d I (W + , Θ + ) < d I (W, Θ).
Thus, after finitely many flips, we may assume that ψ is a birational contraction. As in the proof of Corollary 7.5, the claim follows.
The Lemma above concludes the proof of Theorem 7.1.
Corollary 7.12. Let X be a Q-factorial quasi-projective threefold and let π : X → Z be a birational morphism. Let (F , ∆) be an F-dlt pair on X and suppose F has non-dicritical singularities. Assume that every component of exc(π) is an lc centre for (F , ∆).

terminates.
Proof. By Theorem 7.1, any sequence of flips is eventually disjoint from the lc centres of (F , ∆) and so is eventually disjoint from exc(π), in which case the MMP terminates.

Existence of F-dlt modifications
We now show the existence of an F-dlt modification as in Definition 3.17. The result is a consequence of the existence of flips and special termination and it will be used to prove the base point free theorem in Section 9.
Theorem 8.1 (Existence of F-dlt modifications). Let F be a co-rank one foliation on a Q-factorial projective threefold X. Let (F , ∆) be a foliated pair.
Then (F , ∆) admits a F-dlt modification. Furthermore, if (F , ∆) is lc and Γ = π −1 * ∆ + ǫ(E i )E i , then we may choose π : Y → X so that (1) if Z is an lc centre of (G, Γ) then Z is contained in a codimension one lc centre of (G, Γ), Proof. Let φ : W → X be a sufficiently high foliated log resolution so that every lc centre is contained in a codimension one lc centre. Let H be the foliation induced on W .
We may write We run a (K H + Θ)-MMP over X. By [Spi17, Theorem 8.12] all the required divisorial contractions exists and by Theorem 6.4 all the flips exist. By construction each G i , F j is an lc centre of (H, Θ) and so Corollary 7.12 implies that this MMP terminates. Call this MMP f : W Y and let G be the foliation induced on Y . Note that Lemma 3.8 implies that (G, f * Θ) is F-dlt.
The MMP preserves Q-factoriality and it preserves klt singularities, and so we have that Y is Q-factorial and klt. Denote by π : Y → X the induced morphism. We have K G + f * ∆ = π * (K F + ∆) and so D := f * Θ − f * ∆ is π-nef and π-exceptional. The negativity lemma then implies that f * ∆ − f * Θ = −D ≥ 0. Thus, setting F = −D and noting that f * Θ = π −1 * ∆ + ǫ(E i )E i where we sum over the π-exceptional divisors, we have To see Item (1), we may freely replace (F , ∆) by (G, f * Θ) and so we may assume that (F , ∆) is F-dlt.
Arguing as above we see that it suffices to show that the (K H + Θ)-MMP does not contract any component in the support of G i . By assumption, b i = ǫ(G i ) and so Since K H + ∆ is trivial over X, each step of the (K H + Θ)-MMP is (ǫ(F j ) − a j )F j -negative and so only components in the support of (ǫ(F j ) − a j )F j are contracted. In particular, no component in the support of E i is contracted by the MMP and our result follows.
Theorem 8.2 (Cone theorem for lc pairs). Let X be a normal projective threefold and let F be a co-rank one foliation with non-dicritical singularities. Let (F , ∆) be an lc pair where ∆ ≥ 0 and let H be an ample Q-divisor.
Then there exist countable many curves ξ 1 , ξ 2 , . . . such that Furthermore, for each i, either ξ i is a rational curve tangent to F such that (K F + ∆) · ξ ≥ −6. and if C ⊂ X is a curve such that In particular, there exist only finitely many (K F + ∆ + H)-negative extremal rays.
Proof. By Theorem 8.1, there exists a F-dlt modification π : Y → X for the foliated pair (F , ∆). We may write K G +Γ = π * (K F +∆) where G is the induced foliation on Y and Γ ≥ 0.
Observe that if R ⊂ NE(X) is an extremal ray then there exists an extremal ray R ′ ⊂ NE(Y ) such that π * R ′ = R. Moreover, if R is (K F +∆)-negative then R ′ is (K G +Γ)-negative and so by Theorem 3.19 R ′ is spanned by a rational curve ξ tangent to G with (K G +Γ)·ξ ≥ −6. Then π(ξ) spans R and has all the desired properties.
If C ⊂ X and [C] ∈ R + [ξ i ] for some i then we may apply Lemma 3.18 to conclude that C is tangent to F . 8.1. Contraction in the non-Q-factorial case. Through out this subsection, we assume that (X, Θ) is klt for some Θ ≥ 0 but that X is not necessarily Q-factorial.
Lemma 8.3. Let X be a projective three dimensional variety and let F be a co-rank one foliation on X with non-dicritical singularities. Let ∆ ≥ 0 be a Q-divisor such that (F , ∆) is a log canonical pair, and suppose there exists a small Q-factorialisation π : X → X such that if we write K F + ∆ = π * (K F + ∆), where F is the induced foliation on X, then for any choice of ǫ > 0 we may find Θ such that (1 − ǫ)∆ ≤ Θ ≤ ∆ and (F , Θ) is F-dlt. Let R be a (K F + ∆)-negative extremal ray. Assume that loc(R) = X.
Then there exists a contraction φ R : X → Y of R in the category of algebraic spaces.
Proof. Let H R be a nef Q-Cartier divisor on X which defines a supporting hyperplane for R in NE(X). Let π : X → X be a small Qfactorialization of X as in the hypotheses of the Lemma. First, suppose that loc(R) = D is a divisor. We may find an ǫ > 0 sufficiently small and a Θ as in the hypotheses of the Lemma so that K F + Θ is negative on any extremal ray R ′ in NE(X) such that π * R ′ = R.
Recall that if D contains an irreducible component D 0 transverse to the foliation that D 0 the foliation restricted to D 0 is a P 1 -fibration. In particular, if we let D 0 be the strict transform of D 0 under π we see by adjunction, if F is a general fibre in this P 1 -fibration struction, that We run a K F + Θ-MMP contracting/flipping only π * H R -trivial extremal rays R ′ such that loc(R ′ ) meets the strict transform of D. By Theorem 6.4 we know all the required flips exist. By our above observation we know that if the strict transform of D contains a non-invariant component we may choose this extremal ray to also be K X +Θ-negative.
Thus, by termination of log flips we see that there can only be finitely many flips before each non-invariant component of D is contracted. By Special Termination we see that there are only finitely many such flips before each component of D is contracted.
Denote by Y the step in this MMP after the last component of D is contracted, and let f : X Y denote the induced rational map. Observe that each step of this MMP is π * H R trivial and so π * H R descends to a Q-Cartier divisor M on Y We know that Y is Q-factorial and as in the proof of [Spi17, Lemma 8.15], we know that if S is a divisor on Y then M 2 · S > 0. Moreover, if B is a curve we see that M ·B = 0 if and only if B is the strict transform of a π-exceptional curve, B is the strict transform of a flipped curve or B is the strict transform of a curve C with [C] ∈ R. Notice that there are finitely many such curves and let Σ be the union of all such curves. By [Spi17, Lemma 8.16] there exists a contraction of Σ in the category of algebraic spcaes call it c : Y → Y. Since c contracts every π-exceptional curve and every flipped curve it gives a contraction φ R : X → Y. Now suppose that loc(R) is a curve. In this case we see that if S is a divisor on X then, as in the proof of [Spi17, Lemma 8.15], we have that (π * H R ) 2 · S > 0 and that (π * H R ) · C = 0 if and only if C ⊂ π −1 (loc(R)) ∪ exc(π). As above, we apply [Spi17, Lemma 8.16] to produce a contraction of π −1 (loc(R)) in the category of algebraic spaces, which factors through π.
Theorem 8.4. Let (X, Γ) be a projective three dimensional klt pair and let F be a co-rank one foliation on X with non-dicritical singularities. Let ∆ ≥ 0 be a Q-divisor such that (F , ∆) is a log canonical pair and suppose there exists a small Q-factorialisation π : X → X such that if we write K F +∆ = π * (K F +∆), where F is the induced foliation on X, then for any choice of ǫ > 0 we may find Θ such that (1−ǫ)∆ ≤ Θ ≤ ∆ and (F, Θ) is F-dlt. Let R be a (K F + ∆)-negative extremal ray.
Then the contraction associated to R φ R : X → Y exists in the category of projective varieties and ρ(X/Y ) = 1. In particular, if φ R is birational, then it is extremal.
Proof. First, observe that if loc(R) = X then the result follows directly from [Spi17,Theorem 8.9]. Otherwise, let φ R : X → Y be the contraction onto an algebraic space Y, whose existence is guaranteed by Lemma 8.3. We first show that if M is a Q-Cartier divisor with M · R = 0 then M = φ * R N for some Q-Cartier divisor on Y. Observe that this problem isétale local on Y, so we may freely replace Y by a sufficiently smallétale neighborhood of some point y ∈ Y.
We may approximate every separatrix (formal or otherwise) of F meeting g −1 (y) by global divisors on X. Let S k be the collection of all such divisors. As above we see that For some ǫ > 0 sufficiently small, we may run a (K X + Γ + ǫ(∆ + S k ))-MMP over X, and we obtain a map X X ′ . Let π ′ : X ′ → X be the induced morphism. Each step of this MMP is (K X + Γ)-trivial and so if we let T k be the strict transform of S k and ∆ ′ be the strict transform of ∆ on X ′ , we see that ∆ ′ + T k is nef over X. Observe that T k still approximates the separatrices of the transformed foliation F ′ on X ′ .
Thus, replacing X by X ′ we may freely assume that (∆+ S k )·C ≥ 0 for any π-exceptional curve C. Since K F + ∆ is strictly negative on any φ R -exceptional curve, we see that for 0 < δ ≪ 1 we have is nef over Y. By Lemma 3.15 and the fact that (F, ∆) is log canonical we see that (X, ∆ + S k ) is log canonical and since X is klt we have that (X, (1 − δ)(∆ + S k )) is klt for δ > 0 and so we may apply the base point free theorem to π * M to conclude that there exists a Q-Cartier divisor N on Y with φ * R N = M. Let H R be a nef Q-Cartier divisor on X which defines a supporting hyperplane for R in NE(X). Taking M = H R we see that by applying the Nakai-Moishezon criterion to N that N is ample and hence Y = Y is projective.
Observe that the hypotheses of Theorem 8.4 are satisified if we suppose that (F , ∆) is F-dlt.

Potentially klt varieties.
Definition 8.5. We say that a normal variety X is potentially klt if there exists a Q-divisor ∆ ≥ 0 such that (X, ∆) is klt.
We say that a normal variety X isétale locally potentially klt if for all x ∈ X there is anétale neighborhood U of x such that U is potentially klt.
Lemma 8.6. Let X be a normal projective variety. Suppose that X iś etale locally potentially klt. Then X is potentially klt.
In particular, let φ R : X → Y be the contraction associated to an extremal ray as in Theorem 8.4. Then Y is potentially klt.
Proof. Choose a finiteétale cover Without loss of generality we may assume each g i is Galois, with Galois group G i . Perhaps replacing ∆ i by 1 #G i g∈G i g · ∆ i we may assume that there exists a Zariski open set V i ⊂ X and a Q-divisor is klt and so we may freely assume that g i : U i → X is an open immersion.
There exists m > 0 such that m∆ i ∈ |−mK U i | for all i. Let H be a divisor on X such that O(−mK X + mH) is globally generated. We may assume that, for all i, there exists D i ∈ |−mK X + mH| such that (U i , 1 m D i ) is klt. It follows that, for a general element D ∈ |−mK X + mH|, we have that (X, 1 m D) is klt. Thus, X is potentially klt. To prove our final claim it suffices to check that Y isétale locally potentially klt. So let y ∈ Y and let U be a sufficiently smallétale neighborhood of y, and let X U = X × Y U. By the construction given in proof of Theorem 8.4, there exists a small morphism π : X U → X U and a divisor D ≥ 0 such that (X U , D) is klt and −(K X U +D) is φ R -nef. By the basepoint free theorem, we may find a 0 ≤ A ∼ Q −(K X U + D) such that (X U , D + A) is klt and K X U + D + A is φ R -trivial. Thus, (U, (φ R ) * (D + A)) is klt and so X isétale locally potentially klt.

Base Point Free Theorem
The goal of this section is to prove the base point free theorem. We begin with the following version of the canonical bundle formula: Lemma 9.1. Let X be a normal projective variety with dim(X) ≤ 3 and let (F , ∆) be an lc pair. Suppose that there is a fibration f : X → Y all of whose fibres are tangent to F and such that K F + ∆ ∼ Q,f 0.
Then there is a foliation G on Y such that f −1 G = F , and a Q-divisor Θ ≥ 0 and a semi-ample divisor D such that K F +∆ ∼ Q f * (K G +Θ+D) and (G, Θ) is lc.
Proof. First, notice that since the fibres of f are tangent to F there exists a foliation G on Y so that F = f −1 G. We also have that there exists M on Y so that K F + ∆ ∼ Q f * M.
Consider a commutative diagram as follows where µ, ν are resolutions of singularities. Let F ′ and G ′ be the transformed foliations on X ′ and Y ′ respectively. Write K F ′ + ∆ ′ = ν * (K F + ∆). and so we have We part of the ramification divisor of g and so by [Kol07,Theorem 8.3.7] we may find a nef Q-divisor J and an effective Q-divisor B such that µ * M ∼ Q K G ′ +B +J. Furthermore, by [Amb05] in the case dim(Y ) = 1 and [Kaw97] when dim(Y ) = 2 we know that J is in fact semi-ample.
If B = a i B i , then An explicit calculation shows that Thus, since (F ′ , ∆ ′ ) is lc, it follows that a i ≤ ǫ(B i ) and µ * B ≥ 0. Since J is semi-ample the base locus of µ * J consists of isolated points, and is therefore semi-ample.
Lemma 9.2. Let X be a normal projective threefold and let F be a corank one foliation with non-dicritical singularities. Suppose that (X, D) is klt for some D ≥ 0. Let ∆ = A + B be a Q-divisor such that (F , ∆) is an lc pair, A ≥ 0 is an ample Q-divisor and B ≥ 0. Assume that K F +∆ is not nef, but there exists a Q-divisor H such that K F +∆+H is nef. Let λ = inf{t > 0 | K F + ∆ + tH is nef }. Then there exists a (K F + ∆)-negative extremal ray R such that (K F + ∆ + λH) · ξ = 0, for any ξ ∈ R.
Proof. By Theorem 8.2, there exist only finitely many curves ξ 1 , . . . , ξ m such that A·ξ i > 0 and, if R i = R + [ξ i ] for i = 1, . . . , m, then R 1 , . . . , R m are (K F + ∆)-negative extremal rays. Let It follows easily that µ = 1−λ λ . By construction, there exists j such that 1 λ Thus, the claim follows by taking Lemma 9.3. Let X be a normal projective threefold and let F be a co-rank one foliation with non-dicritical singularities. Let ∆ ≥ 0 be a Q-divisor such that (F , ∆) is a log canonical pair, and suppose there exists a small Q-factorialisation π : X → X such that if we write where F is the induced foliation on X, then for any choice of ǫ > 0 we may find Θ such that Proof. Let Γ = 1 2 A + B and let λ = min{t ≥ 0 | K F + Γ + tA is nef }.
If λ < 1/2 then K F + ∆ is ample and there is nothing to prove. Thus, we may assume that λ = 1/2. By Lemma 9.2, there exists a (K F + Γ)negative extremal ray R such that (K F + ∆) · ξ = 0 for all ξ ∈ R. By Theorem 8.4, there exists a morphism f : X → X ′ which contracts exactly all the curves in R.
Assume first that f is birational. Let F ′ be the foliation induced on X ′ and let A ′ be an ample Q-divisor on X ′ such that A − f * A ′ is also ample. Then there exists a Q-divisor A ′′ ≥ 0 on X ′ and a Q-divisor B ≥ 0 on X such that A ′ ∼ Q A ′′ and if ∆ ′ := f * A ′′ + B ′ , then (F , ∆) is log canonical. Let ∆ ′′ be the image of ∆ ′ in X ′ . Then ∆ ′′ = A ′′ + B ′′ where B ′′ ≥ 0 and (F ′ , ∆ ′′ ) is lc. Note that ρ(X ′ ) < ρ(X). By Lemma 8.6 there exists a Q-divisor D ′ such that (X ′ , D ′ ) is klt.
If f is a flipping contraction then the existence of a small Q-factorialisation π : X ′ → X ′ satisfying the hypotheses of the lemma is an immediate consequence of the existence by such a small Q-factorialisation for X. Thus we may replace (F , ∆) by (F ′ , ∆ ′′ ) and continue. Now suppose f is a divisorial contraction. Consider a diagram as in the proof of Lemma 8.3 where π : X → X is a small Q-factorialisation satisfying the hypotheses of the lemma, f is a K F + Γ-MMP where K F + Γ = π * (K F + Γ) and π ′ is the induced morphism. We claim that π ′ : X ′ → X ′ satisfies the hypotheses of the lemma. It is immediate that X ′ is projective and Q-factorial since X is and π ′ is small. We may choose ǫ, δ > 0 sufficiently small and Θ on X such that (F , Θ) is F-dlt as guaranteed by our hypotheses and such that f is We may therefore replace (F , ∆) by (F ′ , ∆ ′′ ) and continue. After finitely many steps we obtain the claim.
Next assume that f is a fibration. By Lemma 9.1, there is a foliation G on Y , a semi-ample divisor D and Θ ≥ 0 such that (G, Θ) is lc and Notice that, as in the proof of [Spi17, Theorem 8.9], it follows that f is K X -negative. Thus, Y is klt.
Theorem 9.4. Let X be a normal projective threefold and let F be a co-rank one foliation with non-dicritical singularities. Suppose that there exists D ≥ 0 such that (X, D) is klt. Let ∆ be a Q-divisor such that (F , ∆) is a F-dlt pair. Let A ≥ 0 and B ≥ 0 be Q-divisors such that ∆ = A + B and A is ample. Assume that K F + ∆ is nef.
Then K F + ∆ is semi-ample.
Proof. The Theorem follows immediately from Lemma 9.3.

Minimal Model Program with scaling
The goal of this section is to show the existence of a minimal model for a F-dlt pair (F , A + B) where A ≥ 0 is an ample Q-divisor and B ≥ 0. To this end, we are not able to show termination of flips in general, but we can show that a special sequence of flips terminates. This process is called MMP with scaling. Below, we adopt many of the techniques used in [BCHM10].
Let f : X Y be a proper birational map of normal varieties and let D be a Q-divisor on X such that both D and D ′ := f * D are Q-Cartier. We say that f is D-non-positive if for any resolution of indeterminacy p : W → X and q : W → Y , we may write In particular, if (F , ∆) is a F-dlt foliation on a normal projective variety X, then a sequence of (K F +∆)-flips and divisorial contractions is a (K F + ∆)-non-positive birational map. A minimal model of (F , ∆) is a (K F + ∆)-non-positive birational map f : is F-dlt and K F ′ + ∆ ′ is nef.
Lemma 10.1. Let F be a co-rank one foliation on a normal Q-factorial projective threefold X. Let ∆ be a Q-divisor such that (F , ∆) is a F-dlt pair and suppose that F has non-dicritical singularities. Let A ≥ 0 and where α ≥ 0, and D ≥ 0 is a R-divisor whose support is a union of lc centres of (F , ∆).
Then there exists a birational contraction f : X Y which is a minimal model for (F , ∆).
Proof. Let λ = inf{t > 0 | K F + ∆ + tH is nef}. If λ = 0, then K F + ∆ is nef and there is nothing to prove. Otherwise, by Lemma 9.2, there exists a curve ξ in X such that R = R + [ξ] is an extremal ray of NE(X) satisfying: Note that, since (D + αH) · ξ < 0, α ≥ 0 and H · ξ > 0, it follows that ξ is contained in the support of D and, in particular, ξ intersects an lc centre of (F , ∆). By [Spi17, Theorem 8.12] and Theorem 6.4, R defines a divisorial contraction or a flip φ : X X ′ . Let F ′ be the induced foliation on X ′ and let ∆ ′ , H ′ and D ′ be the image in X ′ of ∆, H and D respectively. It follows that K F ′ + ∆ ′ + λH ′ is nef. By Lemma 3.8, (F ′ , ∆ ′ ) is F-dlt.
By Lemma 3.14, there exist Q-divisors A ′ ≥ 0 and B ′ ≥ 0 such that Thus, we may replace X, ∆, F , D, H and α by X ′ , F ′ , ∆ ′ , D ′ , λH ′ and α/λ resepectively and we proceed as above. Theorem 7.1 implies that, after finitely many steps, we obtain a minimal model of (F , ∆).
Lemma 10.2. Let F be a co-rank one foliation on a smooth projective threefold X. Let ∆ = A + B be a Q-divisor such that (F , ∆) is a F-dlt pair, A ≥ 0 is an ample Q-divisor and B ≥ 0. Assume that there exists a Q-divisor D ≥ 0 such that Proof. Note that F admits non-dicritical singularities and that (F , ∆) is F-dlt. We may write D = D 1 + D 2 where D 1 , D 2 ≥ 0 and the components of D 1 are exactly the components of D which are lc centres of (F , ∆). Note that, in particular, D 1 contains all the components of D which are F -invariant. Let k be the number of components of D 2 . We proceed by induction on k.
If k = 0, then D 2 = 0 and the support of D is a union of lc centres of (F , ∆). Let H be a sufficiently ample Q-divisor such that K F + ∆ + H is ample. Then Lemma 10.1 implies that there exists a birational contraction f : X Y which is a minimal model for (F , ∆). We now assume that k > 0. Let Then λ > 0 and K F + ∆ + λD 2 ∼ R D + λD 2 By Item 2, it follows that (F , ∆ + λD 2 ) is F-dlt. By induction, it follows that (F , ∆ + λD 2 ) admits a minimal model X X ′ , which is a birational contraction. Let F ′ be the induced foliation on X ′ and let ∆ ′ , D ′ , D ′ 1 and D ′ 2 be the image of ∆, D, D 1 and D 2 on X ′ respectively. Let H ′ = λD ′ 2 . Then K F ′ + ∆ ′ + H ′ is nef and Thus, Lemma 10.1 implies that there exists a birational contraction X ′ Y which is a minimal model for (F ′ , ∆ ′ ). Let f : X Y be the induced map. Note that f is a birational contraction. In order to show that f : X Y is a minimal model for (F , ∆), it is enough to show that f is (K F + ∆)-non-positive. Let G be the induced foliation on Y and let Γ = f * ∆. By Lemma 3.14, there exists Q-divisors A ′ ≥ 0 and B ′ ≥ 0 such that Γ ∼ Q A ′ + B ′ , A ′ is ample and (F ′ , A ′ + B ′ ) is F-dlt. Thus, Theorem 9.4 implies that K G + Γ is semi-ample.
Let p : W → X and q : W → Y be a resolution of indeterminacy of f . Then, we may write where E, F ≥ 0 are q-exceptional Q-divisors without any common component. Since K G + Γ is semi-ample, it follows that the stable base locus of q * (K G + Γ) + E coincides with the support of E. Let us assume that F = 0. Then, we claim that there exists a component S of F which is contained in the stable base locus of p * D + F . Indeed either there exists a component S of F which is p-exceptional and the claim follows immediately or the image T of a component S of F in X is f -exceptional. In particular, T is contained in the support of D and, by Item 3, T is contained in the stable base locus of D. It follows that S is contained in the stable base locus of p * D + F . Thus, S is a component of E, a contradiction. It follows that F = 0 and, in particular, f is (K F + ∆)-non-positive. Thus, f : X Y is a minimal model for (F , ∆).
Theorem 10.3. Let F be a co-rank one foliation on a Q-factorial projective threefold X. Let ∆ = A + B be a Q-divisor such that (F , ∆) is a F-dlt pair, F has non-dicritical singularities, A ≥ 0 is an ample Q-divisor and B ≥ 0. Assume that there exists a Q-divisor D ≥ 0 such that K F + ∆ ∼ Q D.
Proof. By Lemma 3.13, after possibly replacing A by a Q-equivalent divisor, we may assume that ⌊∆⌋ = 0 and that, for any exceptional divisor E over X, if a(E, F , ∆) = −ǫ(E) then E is invariant and a(E, F ) = a(E, F , ∆) = 0. By [BCHM10, Proposition 3.5.4], we may find a positive integer m and Q-divisors P ≥ 0 and N ≥ 0 such that P + N ∼ Q D and any component of N is contained in the stable base locus of P + N, whilst every component Σ of P is such that mΣ is mobile. Let π : Z → X be a foliated log resolution of (F , ∆ + P + N) which also resolves the base locus of |mΣ| for any component Σ of P . Let G be the induced foliation on Z. We may write K G + ∆ Z = π * (K F + ∆) + F for some Q-divisors ∆ Z , F ≥ 0 without common components. Let C ≥ 0 be a π-exceptional Q-divisor on Z such that π * A − C is ample. Notice that π * A − tC is ample for any 0 < t < 1.
Thus, there exist δ, ǫ > 0 and a Q-divisor Γ ∼ Q ∆ Z − δC + ǫ E i where the sum is taken over all the non-invariant π-exceptional divisors and such that ( (3) we may write where F ′ ≥ 0 is a π-exceptional Q-divisor, whose support contains every exceptional divisor E of π such that a(E, F ) > −ǫ(E), (4) there exists an effective divisor D ′ ∼ Q K G + Γ such that any component of D ′ is either semi-ample or it is contained in the stable base locus of D ′ , and (5) (G, Γ + D ′ ) is a log smooth foliation. Lemma 10.2 implies that (G, Γ) admits a minimal model g : Z Y , which is a birational contraction. We want to show that that the induced map f : X Y is a minimal model of (F , ∆). Let p : W → Z and q : W → Y be proper birational morphisms that resolve the indeterminacy locus of f . Let r : W → X be the induced morphism. Since g is (K G + Γ)-non-positive, we may write where F ′ is the induced foliation on Y , Γ ′ = g * Γ and G ≥ 0 is qexceptional. On the other hand, we also have Since K F ′ + Γ ′ is nef, the negativity lemma implies that G ≥ p * F ′ . In particular, the support of G contains every exceptional divisor E of π such that a(E, F ) > −ǫ(E). Thus, if E ′ is a f −1 -exceptional divisor on Y then E ′ is invariant and a(E ′ , F ) = −ǫ(E) = 0. Moreover, it follows that f is (K F + ∆)-non-positive. Thus, the claim follows.

Existence of F-terminalisations
Theorem 11.1 (Existence of F-terminalisations). Let F be a co-rank one foliation on a Q-factorial threefold X.
Then there exists a birational morphism π : Y → X such that (1) if G is the transformed foliation, then G is F-dlt and canonical (in particular it is terminal along sing(Y )), (2) Y is klt and Q-factorial and Proof. Let µ : W → X be a foliated log resolution of F and let H be the induced foliation on W . Let A be an ample divisor on X and let C ≥ 0 be a µ-exceptional Q-divisor on W such that µ * A − C is ample. Let 0 < δ ≪ 1 such that if Γ = µ * A − δC then where G 1 , G 2 ≥ 0 are µ-exceptional Q-divisors without any common component and the support of G 2 contains all the µ-exceptional divisors with discrepancy greater than zero with respect to F . Note that µ * A− δC is ample. Thus, we may find 0 ≤ A ′ ∼ Q Γ such that (H, A ′ ) is F-dlt. By Theorem 3.19, we may find a sufficiently large positive integer n such that any (K H + A ′ + nµ * A)-negative extremal ray is generated by a curve which is contracted by µ. By choosing n large enough, we may also assume that there exists a Q-divisor D ≥ 0 on W such that is F-dlt and canonical. By Theorem 10.3, K H + A ′′ admits a minimal model f : W Y . If n is sufficiently large, this MMP will only contract µ-exceptional curves and thus, we still have a morphism π : Y → X.
Note that, if G is the induced foliation, then G is F-dlt and canonical and, by Lemma 3.9, it is terminal along sing(Y ). Moreover, Y is klt and Q-factorial. Finally, f * (G 2 − G 1 ) is nef over X and π-exceptional and so the negativity lemma applies to show that f * G 2 = 0. Thus, if E := π * K F − K G then E ≥ 0.
Definition 11.2. We call a modification π : Y → X as in Theorem 11.1 an F-terminalization for the foliated pair (F , ∆).
Theorem 11.3. Let (F , ∆) be a foliated pair on a threefold X. Assume that either Then F has non-dicritical singularities. Furthermore, if (F , ∆) is F-dlt and K X is Q-Cartier then X is klt.
Proof. We will only prove the case where (F , ∆) is canonical. The other one may be handled in a similar manner. Let µ : W → X be a foliated log resolution of (F , ∆) and let H be the induced foliation on W . Our result follows if we can show µ −1 (P ) is tangent to H for all P ∈ X. So suppose for sake of contradiction that there is some P such that µ −1 (P ) is not tangent to H and let C ⊂ µ −1 (P ) be a general curve transverse to the foliation. Write K H + Γ = µ * (K F + ∆) + E where E ≥ 0 is µ-exceptional and Γ ≥ 0, so that E and Γ do not have any common component. Let A ≥ 0 be an ample divisor on X and let G ≥ 0 be a µ-exceptional Q-divisor on W such that µ * A − G is ample. Let F be the sum of all the µ-exceptional non-invariant divisors. There exist sufficiently small ǫ, δ > 0 such that if Θ = µ * A − δG + Γ + ǫF , then we may write where E 1 , E 2 ≥ 0 are µ-exceptional Q-divisors without common components and such that the support of E 2 contains all the µ-exceptional non-invariant divisors.
As in the proof of Theorem 11.1, by Theorem 10.3, we may run a (K H + Θ + nµ * A)-MMP φ : W Y , where n is sufficiently large so that the induced map ν : Y → X is a proper morphism. Let G be the induced foliation on Y . Notice that, the negativity lemma implies that φ * E 2 = 0 and, in particular, φ contracts all the non-invariant µ-exceptional divisors. Moreover, we have Thus, if δ is sufficiently small, then the support of E is contained in the support of E 2 and therefore φ * E = 0. It follows that K G + φ * Γ = K G + ν −1 * ∆ = ν * (K F + ∆) and that every ν-exceptional divisor is G-invariant. Since C is transverse to the foliation we have 0 = φ * C =: C ′ is also transverse to the foliation and so is not contained in any ν-exceptional divisor. Let A 1 and A 2 be two distinct effective Cartier divisors containing ν(C ′ ) = P and write ν * A i = ν −1 On one hand we know that B i · C ′ > 0, on the other hand we know that ν * A i · C = 0 and so ν −1 * A i · C < 0. Let D = ν −1 * A 1 + ν −1 * A 2 and let λ = sup{t > 0|(G, ν −1 * ∆ + tD) is log canonical along C ′ }. Notice that 1 ≥ λ > 0, C ′ is an lc centre of (G, ν −1 * ∆ + tD) and (K G + ν −1 * ∆ + λD) · C ′ < 0. This however is a contradiction of foliation subadjunction, [Spi17,Theorem 4.3] which implies that (K G + ν −1 * ∆ + λD) · C ′ ≥ 0.
To see our final claim, since (F , ∆) is F-dlt, we may find a log resolution µ : W → X only extracting divisors E of discrepancy > −ǫ(E). We run a (K H + µ −1 * ∆+ F )-MMP over X, where F is the sum of all the µ-exceptional non-invariant divisors. Note that this MMP terminates by Corollary 7.12. Let φ : W Y be the output of this MMP, with induced morphism ν : Y → X. Observe that Y is klt. By the negativity lemma, we know that ν is small and so K Y = ν * K X which implies that X is klt.
Remark 11.4. Theorem 11.3 shows that the hypothesis of non-dicriticality in the cone theorem (and in the above results) is superfluous. When X is smooth this result follows from [LPT11, Proposition 3.11].
12. Abundance for c 1 (K F + ∆) = 0 The goal of this section is to prove the following: Theorem 12.1. Let X be a Q-factorial threefold and F be a co-rank one foliation. Let (F , ∆) be a foliated pair with log canonical foliation singularities and such that c 1 (K F + ∆) = 0. Then κ(K F + ∆) = 0.
As we mentioned in the Introduction, the result above is a consequence of [LPT11,Theorem 2] in the case of foliations with canonical singularities defined on a smooth projective variety.
12.1. ∆ = 0 or F is log canonical but not canonical.
Lemma 12.2. Suppose X is a klt surface and F is a rank one foliation on X. Suppose that (F , ∆) is lc and c 1 (K F + ∆) = 0.
Proof. Without loss of generality we may replace (F , ∆) by an F-dlt modification. In particular, we may assume that F has canonical singularities. If ∆ = 0, then our claim follows from [McQ08, Lemma IV.3.1]. If ∆ = 0, then K F is not pseudo-effective. Running an MMP for K F with scaling of some ample divisor, and replacing F by this output we may assume that we have a P 1 -fibration f : X → C such that F is induced by the fibration.
By Lemma 9.1 we see that K F + ∆ ∼ Q f * (K G + Θ) where Θ ≥ 0 is an effective divisor and G is the foliation by points on C. In particular K G = 0 and our result is proven.
Lemma 12.3. Let X be a Q-factorial klt threefold and F be a corank one foliation. Suppose that F is algebraically integrable, F has canonical singularities, (F , ∆) is log canonical and c 1 (K F + ∆) = 0.
Proof. By assumption F admits a meromorphic first integral f : X C where C is a curve. Let µ : X ′ → X be a resolution of indeterminacies of f and let f ′ : X ′ → C be the resolved map. Observe that f ′ is a holomorphic first integral of F ′ , the transform of F on X ′ . As F has canonical singularities, hence non-dicritical singularities by Theorem 11.3, if p ∈ X then µ −1 (p) is tangent to F ′ . Since f ′ is a holomorphic first integral of F ′ this implies that f ′ (µ −1 (p)) is a single point and so f ′ contracts every fibre of µ. The rigidity lemma then implies that in fact f : X → C is a morphism.
We apply Lemma 9.1 to write K F + ∆ ∼ Q f * Θ for some Q-divisor Θ ≥ 0 and we can conclude.
Proposition 12.4. Let X be a Q-factorial threefold and F be a co-rank one foliation. Let (F , ∆) be a foliated pair with log canonical foliation singularities. Suppose that c 1 (K F + ∆) = 0 and that either ∆ = 0 or F is log canonical but not canonical.
Proof. By Lemma 12.3 we may assume that F is not algebraically integrable. By Theorem 11.1, we may replace (F , ∆) with a F-terminalization, so we may assume without loss of generality that (F , ∆) is F-dlt and canonical and that ∆ = 0. In particular, we may assume that X is klt. In this case, K F is not pseudo-effective and so there exists a diagram W X B p q where q : W → B parametrizes a dominant family of rational curves tangent to F . Suppose that dim(B) = 2. Let G be an ample divisor on B. Since F has canonical singularities, hence non-dicritical singularities by Theorem 11.3, we see that if p contracts a curve ξ then q must also contract ξ. In particular, we see that if F is a general fibre of q, then p(F ) · p * q * G = 0.
Let A be an ample divisor on X. Given a sufficiently large positive integer m, we may run a K X -MMP with scaling of A + p * q * (mG) Denote this MMP by φ : X X ′ , and let (F ′ , ∆ ′ ) be the induced foliated pair on X ′ . Each step of this MMP is (K F + ∆)-trivial and so we see that (F ′ , ∆ ′ ) is lc, c 1 (K F ′ + ∆ ′ ) = 0 and that κ(K F ′ + ∆ ′ ) = 0 implies that κ(K F + ∆) = 0.
Observe that K X ·p(F ) < 0 and so we may choose m sufficiently large so that this MMP must terminate in a Mori fibre space f : X ′ → S such that S is a surface and f contracts the strict transform of p(F ). Thus, the fibration f is tangent to F ′ . By Lemma 9.1, there is a foliation G on S so that F ′ = f −1 G and a semi-ample divisor D and divisor Θ ≥ 0 such that K F ′ + ∆ ′ ∼ Q f * (K G + Θ + D) and (G, Θ) is lc. Observe that since F is not algebraically integrable we know that D = 0. Since f is K X ′ -negative we see that, in addition, S is klt. Thus, we can apply Lemma 12.2 to conclude that K G + Θ, and hence K F ′ + ∆ ′ , is torsion. Now suppose that dim(B) ≥ 3. If b ∈ B is a general point, then there exists a closed subset where W x denotes the fibre of f over x. By assumption dim(Z b ) ≥ 1. By non-dicriticality, it follows that for any b ′ ∈ Z b , p(W b ) and p(W b ′ ) are generically contained in the same leaf of F , and so p(W × B Z b ) is generically contained in the leaf containing p(W b ). This implies that the leaf containing p(W b ), and hence the general leaf, is algebraic, in which case we conclude by Lemma 12.3. 12.2. ∆ = 0 and F is canonical. In this section F is a co-rank one foliation on a threefold with c 1 (K F ) = 0. Suppose that F has canonical singularities and X is Q-factorial.
Lemma 12.5. We may freely replace F by an F-terminalization. Thus we may assume that X is klt and F is terminal along sing(X), in particular, sing(X) is tangent to F . Proof. Let π : Y → X be an F-terminalization, whose existence is guaranteed by Theorem 11.1 and let G be the induced foliation on Y . By definition, K G + F = π * K F where F ≥ 0. On the other hand, since F is canonical, K G = π * K F + E where E ≥ 0. Thus E = F = 0 and so c 1 (K G ) = 0. Furthermore, if K G is torsion then so is K F . Lemma 12.6. Let φ : X X ′ be a sequence of steps of a K X -MMP and let F ′ be the transformed foliation. Then (2) X ′ has klt and Q-factorial singularities, (3) F ′ has canonical singularities, and (4) sing(X ′ ) is tangent to F ′ . Moreover, if K F ′ is torsion then so is K F .
Proof. Each step of the MMP is K F -trivial so we see that F ′ has canonical singularities and c 1 (K F ′ ) = 0. Furthermore, we see that K F = φ * K F ′ and so if K F ′ is torsion then so is K F . Since X has klt and Q-factorial singularities it follows that X ′ has klt and Q-factorial singularities.
By Theorem 11.3 at each step of the MMP the transformed foliation has non-dicritical singularities and so we see that only curves tangent to F are contracted by the MMP. In particular, we see that the flipping and flipped loci are all tangent to the foliation.
To prove Item (4), consider a step in the K X -MMP, call it f : Y W and let F Y and F W be the induced foliations on Y and W respectively. We claim that if f is a divisorial contraction, then exc(f ) is foliation invariant. Indeed suppose not. By our above observation, f contracts a divisor E transverse to the foliation to a curve C and such that the foliation restricted to E must be tangent to the fibration E → C. Let F be a general fibre of E → C. By Lemma 3.15 we know that Thus, all divisorial contractions in the MMP only contract invariant divisors and so by Lemma 12.5 we may conclude that sing(X ′ ) is indeed tangent to F ′ .
Proof. The proof of [LPT11, Theorem 3.7] works equally well in the case where X is singular.
Lemma 12.8. Suppose we have a morphism f : X → S where S is a surface with klt singularities. Suppose furthermore that K F ∼ Q,f 0 and the fibres of f are tangent to F .
Proof. The argument used in Proposition 12.4 applies here.
Lemma 12.9. Suppose we have a morphism f : X → S where S is a surface with klt singularities and κ(S) ≥ 0. Suppose moroever that the fibres of f are generically transverse to F Then κ(K F ) = 0.
Proof. The pullback of a pluri-canonical form on S restricts to a nonzero form on the leaves of F , i.e., we have a non-zero map for all m ≥ 0. By assumption H 0 (S, mK S ) = 0 for some m sufficiently divisible, and our result follows.
We will need the following definition and result found in [Tou16].
Definition 12.10. Let X be a projective manifold and let F be a corank one foliation on X. Let H 1 , ..., H p be F -invariant hypersurfaces. We say that F is of KLT type with respect to H 1 , ..., H p if there exist rational numbers 0 ≤ a i < 1 such that Theorem 12.11. Let X be a projective manifold and let F be a corank one foliation on X. Let H 1 , ..., H p be F -invariant hypersurfaces. Suppose that F is of KLT type with respect to H 1 , ..., H p Then either Proof. This is [Tou16, Theorem 6] and [Tou16, Theorem 9.7].
We will also need the following classification theorem on surface foliations: Theorem 12.12. Let X be a normal projective surface and let L be a rank one foliation on X with canonical foliation singularities. Suppose c 1 (K L ) = 0.
Then there exists a finite cover τ : X → X and a birational morphism µ : X → Y such that if L and G are the induced foliations onX and Y respectively, then τ is ramified along L-invariant divisors and µ contracts K L -trivial rational curves tangent to L. Moreover, one of the following holds: (1) X = C × E/G where g(E) = 1, C is a smooth projective curve, G is a finite group acting on C ×E and G is the foliation induced by the G-invariant fibration C × E → C; (2) G is a linear foliation on the abelian surface Y ; (3) Y is a P 1 -bundle over an elliptic curve and G is transverse to the bundle structure and leaves at least one section invariant; or (4) Y ∼ = P 2 , P 1 × P 1 or F n (the n-th Hirzebruch surface) and G admits at least 3 invariant rational curves. If Y ∼ = P 1 × P 1 or F n then at least 2 of these invariant curves must be fibres of a P 1 -bundle structure on Y .
Proof. We have an equality of divisors K F + N * F = K X , in particular we see that c 1 (N * F ) = c 1 (K X ). If κ(X) = 3, then κ(N * F ) = 3, a contradiction of the Bogomolov-Castelnuovo-De Franchis inequality (see [GKKP11,Theorem 7.2] for the proof of this statement in the singular setting). Thus, we may assume that κ(X) ≤ 2. Moreover, by Lemma 12.6 we may assume that K X is nef.
We distinguish two cases. We first assume that κ(N * F ) = −∞. Let µ : Y → X be a resolution of singularities of X and let E be the reduced divisor whose support coincides with the µ-exceptional divisor and G be the transformed foliation. By Lemma 12.6 Item (4), sing(X) is tangent to F and so E is and is µ-exceptional. We claim that ⌊E 1 ⌋ = 0. Indeed, this is a local problem on both X and Y , so perhaps shrinking both we may assume that E 1 = aE for some rational number a > 0 so that E 1 consists of a single component and N * where σ is the index one cover associated to N * F . Let F ′ be the pull back of F along σ and let G ′ be the pull back of G along τ . Assume that the ramification index along E is r. Since X ′ is klt, by [GKKP11] we have a morphism ν * (Ω Thus, −ra + (r − 1) = c ≥ 0 which implies that a ≤ r−1 r < 1, as claimed.
Since N * G + E 1 is pseudo-effective and ⌊E 1 ⌋ = 0, it follows that G is a KLT type foliation.
Let f : X → C be the Iitaka fibration associated to K X . If F is the foliation induced by f then we may conclude by Lemma 12.3. Otherwise, the general fibre of f is transverse to F . Let X p be the fibre over p ∈ C, and let F p be the foliation restricted to X p . For general p we see that c 1 (K Fp ) = 0 and that X p is not uniruled. We claim that either F p is algebraically integrable or N * Fp is torsion for general p. Indeed, let τ : Y p → X p and µ : Y p → Z p be the cover and the birational contraction, as in Theorem 12.12. Let G p and H p be the foliation induced on Y p and Z p respectively. Note that µ contracts only invariant K Gp -trivial curves.
Since X p is not uniruled, then the same is true for Y p and so we see that (Z p , H p ) falls into either Case (1) or Case (2). In Case (1), F p is algebraically integrable. In Case (2), observe that H p is smooth (hence terminal) and so µ must be the identity. Moreover, either G (and thus also F p ) is algebraically integrable, or it has no invariant curves. In the latter case we see that τ isétale in codimension 1 and so τ * N * Fp = N * Gp . However, N * Gp is trivial, hence N * Fp is torsion. If F p is algebraically integrable for general p then so is F and we are done by Lemma 12.3 and so we are free to assume that N * Fp is torsion for general p.
Thus, we may find a Q-divisor A on X whose support is contained in fibres of f , but contains no fibre of f and a divisor B on C such that f * B ∼ Q N * F + A. However, K X ∼ Q f * H where H is an ample divisor on C and K X ≡ N * F . Thus, A = 0 and B ≡ H. In particular, B is ample and κ(N * F ) = 1. It follows, by Lemma 12.14 below, that F is algebraically integrable and so we may conclude by Lemma 12.3.
We now assume that κ(N * F ) ≥ 0, and so N * F ∼ Q D ≥ 0. If κ(X) = 0, then since K X is nef, it follows that both N * F and K X , hence K F are torsion, and we are done.
Suppose κ(X) ≥ 1. Let f : X → B be the canonical map. By assumption N * F ∼ Q D ≥ 0 and D is numerically equivalent to f * H where H ≥ 0 is ample. However, this implies that D is actually supported on fibres of f , and is in fact equal to a sum of fibres (with the appropriate multiplicities), and so D = f * B for some ample divisor B. But this implies that κ(N * F ) ≥ 1. By Lemma 12.14 below, it follows that F is in fact algebraically integrable and so we conclude by Lemma 12.3.
Lemma 12.14. Let X be a klt variety and let F be a co-rank one foliation on X. Suppose that κ(N * F ) = 1. Then F is algebraically integrable.
Proof. Perhaps replacing X by a cover we may assume without loss of generality that we may find ω 1 , ω 2 ∈ H 0 (X, O X (N * F )) ⊂ H 0 (X, Ω [1] X ) both non-zero such that ω 1 = f ω 2 where f is a non-constant meromorphic function on X.
Let µ : X → X be a resolution of singularities. By [GKKP11], ω i lifts to a global 1-form ω i on X, for i = 1, 2. Thus, ω i and hence ω i are closed. Observe that 0 = dω 1 = d(f ω 2 ) = df ∧ ω 2 and so f is a meromorphic first integral of F as required.
Lemma 12.15. Let π : X → B be a fibration over a curve. Suppose that X has klt singularities. Let L be a rank one foliation on X tangent to the fibres of X → B. Suppose that c 1 (K L ) = 0 and that L has canonical singularities above the generic point of B. Suppose furthermore that if L is singular above the generic point of B then ρ(X p ) ≥ 2 where X p is the fibre over a general point p ∈ B.
Then either L is algebraically integrable or κ(K L ) = 0.
Proof. For p ∈ B let L p be the foliation restricted to X p . We have where ∆ p ≥ 0. For general p we know that ∆ p = 0, otherwise L would be uniruled, a contradiction. By Theorem 12.12, for a general fibre X p , we may find a finite morphism τ p : X p → X p , ramified along foliation invariant divisors such that L p is generated by a global vector field. Thus, we may find a finite morphism τ : X → X such that τ agrees with τ p along a general fibre, and τ is ramified along invariant divisors and finitely many fibres. Let L be the induced foliation on X. Since all the fibres of X → B are L-invariant we have that K L = τ * K L . Thus, we may freely replace (X, L) by ( X, L).
We argue based on which case the general fibre in falls into in Theorem 12.12.
In Case (1), we see that L is algebraically integrable and so we are done.
In Case (2), we have that X p is an abelian surface for general p and L p is a linear foliation on the abelian surface.
Observe that since L| Xp ∼ 0 and c 1 (L) = 0, there exists a Cartier divisor L on B such that L = π * L and c 1 (L) = 0. Let B 0 ⊂ B be an open set over which X → B is smooth. Perhaps shrinking B 0 we may assume that L is smooth and L p is a linear foliation on the abelian surface X p for all p ∈ B 0 and that B 0 is affine.
In Case (3), we run a K X -MMP over B, call it φ : X X ′ and let L ′ be the foliation induced on X ′ . Each step of this MMP is L-trivial. Thus, it suffices to check that κ(K L ′ ) = 0. A general fibre is uniruled, but not rationally connected, and so this MMP terminates in a Mori fibre space g : X ′ → S over B with dim(S) = 2. Let h : S → B be the induced morphism. As in the proof of Lemma 12.6 we see that φ only contracts curves tangent to L.
By assumption, we know that the fibres of g are generically not tangent to L and so we have a generically surjective sheaf morphism dg : (g * L) * * → T S/B . For generic p ∈ B, let X ′ p (resp. S p ) be the fibre over p of g : X ′ → B (resp. h : S → B) and let Σ p denote one of the L ′ p -invariant sections of the P 1 -fibration X ′ p → S p , where L ′ p is the foliation restricted to X ′ p . Then 0 = K L ′ p · Σ p = c 1 (K Σp + Z) where Z ≥ 0 is the contribution to the different from sing(L ′ p ). However, K Σp = 0 and so Z = 0 implying that L ′ p is smooth along Σ. Thus, we see that dg : (g * L) * * | Sp → T S/B | Sp is in fact surjective.
Since ρ(X ′ /S) = 1 and K L is g-trivial we see that (g * L) * * is a rank one reflexive sheaf and c 1 ((g * L) * * ) = 0. Thus, we have (g * L) * * = T S/B (− a i F i ) where F i are supported on fibres of S → B.
Since c 1 (T S/B (− a i F i )) = 0 and noting that T * S/B = K S/B − R where R is the ramification divisor of h, we have Thus, for some Q-divisor M on B such that c 1 (M) = 0. We apply [Amb05, Theorem 3.5] to conclude that M ∼ Q 0 and hence K L is torsion. In Case (4), we proceed in a similar fashion to the above case with a few modifications. For general p let D 0,p and D ∞,p denote two L p invariant divisors which are fibres in a P 1 -fibration structure on X p .
First, assume that there exist two divisors D 0 and D ∞ on X such that D 0 ∩ X p = D 0,p and D ∞ ∩ X p = D ∞,p for general p and that if we run K X -MMP φ : X X ′ over B we terminate in a Mori fibre space g : X ′ → S where S is a surface, D 0 and D ∞ are not contracted by φ and g contracts the strict transforms of D 0 and D p , call them D ′ 0 and D ′ ∞ . Arguing as above we see that c 1 ((g * L) * * ) = T S/B (−Σ 0 − Σ ∞ − a i F i ) where Σ 0 = g(D ′ 0 ) and Σ ∞ = g(D ′ ∞ ) are reduced divisors dominating B and F i are supported on fibres of S → B.
Since c 1 (T S/B (−Σ 0 − Σ ∞ − a i F i )) = 0 and noting that T * S/B = K S/B − R where R is the ramification divisor of h we have c 1 (K S/B + Σ 0 + Σ ∞ + a i F i − R) = 0. Thus We again apply [Amb05, Theorem 3.5] to conclude that M ∼ Q 0 and hence K L is torsion. Thus to conclude it suffices to arrange the existence of D 0 and D ∞ and such a Mori fibre space structure. Let p ∈ B be a general point. Observe that L is singular and so by assumption ρ(X p ) ≥ 2. Then we may find a sufficiently smallétale neighborhood U of p such that X × B U admits divisors D 0 and D ∞ as required and an MMP over U terminating in the desired Mori fibre space structure. Thus, we may find a (possibly ramified) cover B → B such that X = X × B B admits such an MMP over B. Let L be the pulled back foliation. Observe that σ : X → X is ramified only along L-invariant divisors and so K L = σ * K L and thus we may freely replace (X, L) by (X, L) and our result follows.
In fact, if L is algebraically integrable a similar argument to Lemma 12.3 shows that κ(K L ) = 0.
Corollary 12.16. Suppose we have a morphism f : X → C where C is a smooth curve of positive genus.
Proof. By Lemma 12.3, we may assume that F is generically transverse to f . Let L be the foliation in curves tangent to both F and the fibration f : X → C. We have an exact sequence where Z is supported on the components of fibres which are F -invariant and on subvarieties of codimension at least 2. Thus, we have K F = K L + f * K C + D where D ≥ 0 is the codimension one part of Z. By Lemma 12.7, F is not uniruled and so we know that K L is pseudoeffective. By assumption, f * K C is nef, and since K F is numerically trivial we must have C is genus one and D = 0. So K L ∼ Q K F , in particular, it suffices to prove that K L is torsion.
Observe that if L is algebraically integrable, then so is F in which case we are done by Lemma 12.3. Thus, we may assume that L is not algebraically integrable.
If L is singular above the generic point of C, we claim that we may freely assume that ρ(X p ) ≥ 2 for general p ∈ B. Assuming the claim we apply Lemma 12.15 to conclude.
To prove the claim observe that F is singular above the generic point of C. Let Σ ⊂ sing(F ) be a component dominating C and let µ : Y → X be a blow up along Σ followed by a foliated log resolution. Let E be the strict transform of the exceptional divisor of the blow-up in Y and let G be the induced foliation on Y . Note that a(E, F ) = 0. Pick an ample divisor A on X and let G be the reduced sum of all the µ-exceptional divisors. Then, as in the proof of Theorem 11.1, we may find a sufficiently large positive integer m and a sufficiently small δ > 0 so that if φ : Y W is the result of a (K G + µ * (nA) − δG)-MMP, whose existence is guaranteed by Theorem 10.3, then the induced map π : Y → X is a proper morphism and φ contracts exactly all the µexceptional divisors of positive discrepancy with respect to F . Let H be the foliation induced on W . It follows that c 1 (K H ) = 0 and that π extracts a divisor dominating Σ. Thus, replacing X and F by W and H respectively, we may assume that ρ(X p ) ≥ 2 for general p ∈ B.
Proof. First, assume that K X is pseudo-effective. Then κ(K X ) ≥ 0 and we apply Lemma 12.13 to conclude.