On the relative Minimal Model Program for threefolds in low characteristics
Christopher Hacon Jakub Witaszek



We show the validity of the relative dlt MMP over -factorial threefolds in all characteristics p>0. As a corollary, we generalise many recent results to low characteristics including: W-rationality of klt singularities, inversion of adjunction,  and normality of divisorial centres up to a universal homeomorphism.


[2010]14E30, 14J17, 13A35


Regularized pseudopotential for mean-field calculations
    K Bennaceur^1, J Dobaczewski^2,3,4, T Haverinen^3,5 and M Kortelainen^3,5
    
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§ INTRODUCTION



One of the fundamental tools used in the study of algebraic varieties is the Minimal Model Program (MMP), which is a higher-dimensional analogue of the classification of surfaces. This program has been a topic of intense research in the last decades and a major part of it is known to hold for projective varieties defined over a field of characteristic zero (see <cit.>).    


Moreover, in recent years, there has been substantial progress in the MMP in positive characteristic. In particular, it has been shown that the program is valid for surfaces over excellent base schemes (see <cit.>) and for three-dimensional varieties defined over perfect fields of characteristic p>5 (see <cit.>). This has led to many striking applications, for example, Gongyo, Nakamura, and Tanaka used the MMP to show the existence of rational points on log Fano threefolds defined over finite fields 𝔽_p^n for p>5.

The goal of this article is to extend some of the foundational results on the MMP for threefolds, and applications thereof, to low characteristics. Our main result is the following.

  Let (Y,Δ) be a three-dimensional -factorial dlt pair defined over a perfect field of characteristic p>0. Assume that there exists a projective birational morphism π Y → X over a normal -factorial variety X such that (π) ⊆⌊Δ⌋. Then π-relative contractions and flips exist for K_Y+Δ, and we can run a (K_Y+Δ)-MMP over X which terminates with a minimal model.

Theorem <ref> allows us to study  three-dimensional singularities by means of the MMP. In particular, it can be used to construct Kollár components. 

The proof of the existence of flips for threefolds in positive characteristic p>5 relies on the fact that pl-flipping contractions are purely relatively F-regular. Although this is not true in general when p ≤ 5, we show that pl-flipping contractions occurring in the relative dlt-MMP over -factorial threefolds, as in the theorem above, do satisfy this property. Therefore, we can make use of the same strategy as in <cit.> to conclude the construction of such pl-flips.

In order to show the existence of pl-contractions, it was proven in <cit.> that divisorial centres of three-dimensional plt pairs are normal in characteristic p>5, so the main theorem of <cit.> could be invoked. However, this result about divisorial centres is false in characteristic  p=2, in general, as shown in <cit.> (cf. <cit.>). Nevertheless, Gongyo, Nakamura, and Tanaka observed that in order to use <cit.>, it is enough to show that divisorial centres are normal up to a universal homeomorphism; moreover, they proved that this property of divisorial centres is stable under the MMP. Therefore, they were able to show that the existence of pl-flips in low characteristic implies the existence of minimal models (cf. <cit.>). The same strategy concludes the proof of Theorem <ref> and implies the validity of the following results (cf. <cit.>).

  Let (X,S+B) be a -factorial three-dimensional plt pair defined over a perfect field of characteristic p>0, where S is an irreducible divisor. Then the normalisation f S̃→ S is a universal homeomorphism.

In fact, we show that all log canonical centres of -factorial three-dimensional dlt pairs are normal up to a universal homeomorphism (see Remark <ref>).

Furthermore, we obtain the following applications of Theorem <ref>.
  Let X be a klt three-dimensional -factorial  variety defined over a perfect field of characteristic p>0. Then X has W-rational singularities.


  Let (X,Δ) be a three-dimensional -factorial  log pair defined over a perfect field of characteristic p>0. Then a dlt modification of (X,Δ) exists, that is, a birational morphism π Y → X such that (Y, π^-1_*Δ + (π)) is dlt, -factorial, and minimal over X.
 

  Consider a -factorial three-dimensional log pair (X,S+B) defined over a perfect field of characteristic p>0, where S is an irreducible divisor. Then (X,S+B) is plt on a neighborhood of S if and only if (S̃,B_S̃) is klt, where S̃ is the normalisation of S and B_S̃ is the different.
 


<cit.> to low characteristics, we give a completely new proof of rational chain connectedness of three-dimensional log Fano varieties (see <cit.>), one that is valid in characteristic p>2. The proof does not make use of the MMP for threefolds and is independent of the above results.
  Let (X,Δ) be a projective geometrically -factorial three-dimensional klt pair defined over a perfect field of characteristic p>2 such that -(K_X+Δ) is ample. Assume that its base change to some uncountable algebraically closed field is ℚ-factorial. Then (X,Δ) is rationally chain connected.

The assumption on -factoriality can likely be weakened. To prove this result, it suffices to show that if ϕ X  Z is the MRCC (maximally-rationally-chain-connected) fibration, then Z = 0. When Z >0, we argue that Z is uniruled which yields a contradiction. To this end, we split the proof into three cases. If Z = 1, then ϕ is proper and the rational chain connectedness of Z follows from the cone theorem (see <cit.>). When Z = 2, we resolve ϕ and apply the canonical bundle formula from <cit.>. If Z = 3, we get that Z is uniruled by bend-and-break.

Theorem <ref> and Theorem <ref> imply that the main result of <cit.> holds for p>2.
  Let (X,Δ) be a projective three-dimensional geometrically -factorial klt pair defined over a finite field 𝔽_p^n of characteristic p>2 such that -(K_X+Δ) is ample. Assume that its base change to some uncountable algebraically closed field is ℚ-factorial. Then the number of 𝔽_p^n-rational points satisfies 
    |X(𝔽_p^n)| ≡ 1  (mod p^n).





Moreover, using Theorem <ref>, we may generalise some other results previously known only when p>5.

	
  * divisorial contractions of extremal rays exist in the category of -factorial three-dimensional varieties for p>0,
	
  * the tame fundamental group of three-dimensional -factorial klt singularities is finite when p>0, and
	
  * minimal log canonical centres of -factorial lc pairs (X,Δ), with X being klt, are normal up to a universal homeomorphism for p>0.


Lastly, we show the validity of the (K_X+Δ)-MMP in the setting of dlt reductions. In particular, this allows for the construction of canonical skeletons of degenerations of varieties with non-negative Kodaira dimension (see <cit.>). Here R is a local ring of a curve C defined over a perfect field k of characteristic p>0. We denote the special point s ∈ R and the generic point η∈ R.  Let (𝒳, Φ) be a -factorial dlt pair of dimension three and projective over C and let  (X,Δ):=(𝒳, Φ)× _C  Spec(R).

  Let (X,Δ) be a  dlt pair as above so that in particular X is three-dimensional -factorial projective over R. Let ϕ X → R be the natural projection and suppose that ϕ^-1(s) ⊆⌊Δ⌋. Then we can run a (K_X+Δ)-MMP over R which terminates with a minimal model or a Mori fibre space.
 

This article is organised as follows. In Section 2 we gather preliminary results on F-regularity, the MMP in positive characteristic, and W-rationality. In Section 3 we show Theorem <ref>, Corollary <ref>, Corollary <ref>, and Corollary <ref>. In Section 4 we show Theorem <ref>. Section 6 pertains to the study of other applications of the main theorem.



§ PRELIMINARIES

A scheme X will be called a variety if it is integral, separated, and of finite type over a field k. Throught this paper, k is a perfect field of characteristic p>0. We refer to <cit.> for basic definitions in birational geometry, to <cit.> for the theory of nef reduction maps and nef dimension, and to <cit.> for results pertaining to MRCC (maximally-rationally-chain-connected) fibrations. 
We remark that in this paper, unless otherwise stated, if (X,B) is a pair, then B is a ℚ-divisor.


We advise the reader to consult <cit.> in regards to the subtleties of birational geometry over a perfect field. Let us just recall that being klt, plt, lc, or normal is preserved under taking a base change to an uncountable algebraic closure. Similarly for the ampleness, semi-amplenes, nefness and bigness of a line bundle. The same also holds for rational chain connectedness (see <cit.>). However, this is not the case with -factoriality.  



 §.§ Relative F-regularity

The study of relative F-regularity plays a vital role in the proof of the existence of flips for threefolds in characteristic p>5, acting as a replacement for vanishing theorems (see <cit.>). 

For the convenience of the reader, we recall the basic definitions and results. This section is based on <cit.>.
 For an F-finite scheme X of characteristic p>0 and an effective -divisor Δ, we say that (X,Δ) is globally F-split if for every e ∈_>0, the natural morphism

    _X → F^e_* _X(⌊(p^e-1)Δ⌋)

splits in the category of sheaves of _X-modules.

We say that (X,Δ) is globally F-regular if for every effective divisor D on X and every big enough e ∈_>0, the natural morphism

    _X → F^e_* _X(⌊(p^e-1)Δ⌋ + D)

splits.

We say that (X,Δ) is purely globally F-regular, if the definition above holds for those D which intersect ⌊Δ⌋ properly (see <cit.>*Definition 2.3(2)).

The local versions of the above notions are called F-purity, strong F-regularity, and pure F-regularity, respectively. Moreover, given a morphism f  X → Y, we say that (X,Δ) is F-split, F-regular, and purely F-regular over Y, if the corresponding splittings hold locally over Y (see <cit.>*Definition 2.6).

Let us recall that the relative inversion of F-adjunction holds.
 
Let (X,S+B) be a plt pair where S is a prime divisor, and let f  X → Z be a proper birational morphism between normal varieties  defined over a perfect field of characteristic p>0. Assume that -(K_X+S+B) is f-ample and (S̅,B_S̅) is globally F-regular over f(S), where S̅ is the normalization of S, and B_S̅ is defined by adjunction K_S̅ + B_S̅ = (K_X + S + B)|_S̅. Then (X,S+B) is purely globally F-regular over a Zariski-open neighbourhood of f(S) ⊆ Z. 

Furthermore, divisorial centres of purely F-regular pairs are normal.
   Let (X,S+B) be a purely F-regular pair defined over a perfect field k of characteristic p>0 where S is a prime divisor. Then S is normal.


This follows from <cit.> by taking a base change to the algebraic closure of k.




In order to deal with non--factorial singularities, we also need a relative version of the above result.
  Let (X,S) be a plt three-dimensional pair defined over an algebraically closed field of characteristic p>5, and let f  X → Z be a proper birational morphism between normal varieties such that f(S)=2. If -(K_X+S) is f-ample, then Z is strongly F-regular along f(S).

This does not follow from Proposition <ref> directly, because K_Z + f(S) is almost never -Cartier. Further, let us note that we will need to apply Proposition 
<ref> in the case when X is not -factorial.

As above, S is normal. The normalization morphism f(S)^ν→ f(S) is affine, and so, by <cit.>*Theorem 3.1, the pair (S, Δ_diff), defined by adjunction K_S + Δ_diff = (K_X + S)|_S, is globally F-regular over f(S) (see Remark <ref>). Lemma <ref> and <cit.>*Lemma 2.12 imply that Z is strongly F-regular along f(S).


In the proposition above, we used a relative version of inversion of adjunction, which is a consequence of <cit.>*Theorem B. The proof is very similar to <cit.>*Corollary 5.4 and <cit.>*Lemma 2.7, but we include it here for the convenience of the reader.




 §.§ MMP for threefolds in arbitrary positive characteristic

In this subsection we gather some results on the Minimal Model Program for threefolds which are valid in any characteristic p>0.

Firstly, the cone theorem is known for pseudo-effective adjoint divisors.

 
Let (X,Δ) be a three-dimensional -factorial log pair, projective over a perfect field k. Assume that K_X+Δ is pseudo-effective. Then there exists a countable collection of curves {C_i} such that 


  * NE_1(X) = NE_1(X) ∩ (K_X+Δ)_≥ 0 + ∑_i _≥ 0[C_i].

  * All but a finite number of the C_i are rational and satisfy 0 < -(K_X+Δ) · C_i ≤ 3.

  * The collection of rays {[C_i]} does not accumulate in the halfspace (K_X+Δ)_<0.



 

Let (X,Δ) be a three-dimensional -factorial log canonical pair which is projective over a separated scheme U of finite type over a perfect field k of characteristic p>0. Assume that K_X+Δ is pseudo-effective over U. Then, for every ample -divisor A, there exist finitely many curves C_1, …, C_r such that

    NE(X/U) = NE(X/U)_K_X+Δ+A≥ 0 + ∑_i=1^r _≥ 0[C_i].

 


Secondly, a variant of the base point free theorem for threefolds is known in which the contraction is a map onto an algebraic space (see <cit.>, cf. <cit.>) and in fact the full base point free theorem for threefolds is valid when k=.
 We say that a dlt pair (X,Δ) has normal divisorial centres up to a universal homeomorphism if for every irreducible divisor D ⊆⌊Δ⌋, the normalisation f D̃→ D is a universal homeomorphism.


  Let (X,Δ) be a three-dimensional -factorial dlt pair admitting a projective morphism π X → U to a  scheme U defined over a perfect field of characteristic p>0. Assume that (X,Δ) has normal divisorial centres up to a universal homeomorphism, and let R be an extremal ray of K_X+Δ over U such that R · S < 0 for an irreducible divisor S ⊆⌊Δ⌋. Then the contraction X→ Z of R exists as a projective morphism of normal varieties,  and ρ(X/Z)=1.


Clearly, R ⊆ S. Pick an ample -divisor H such that K_X+S+B+H is nef and

    NE_1(X) ∩ (K_X+S+B+H)^⊥ = _≥ 0[R].
 
Then 𝔼(L) ⊆ S, and so it is enough to show that (K_X+S+B+H)|_S is semiample. Let g S̃→ S be the normalisation of S. Then (K_X+S+B+H)|_S̃ = K_S̃ + Diff, where Diff is the different. Since (X,S+B) is plt, (S̃,Diff) is klt, and K_S̃ + Diff is semiample by the base point free theorem for surfaces (see <cit.>). As g is a universal homeomorphism, (K_X+S+B)|_S is semiample (see Lemma <ref>). 




This result is based on the following theorem of Keel.
 
Let L be a nef line bundle on a scheme X defined over a field k of characteristic p>0. Then L is semiample if and only if so is L|_𝔼(L).

Here, 𝔼(L) is the closure of the union of all irreducible subvarieties Z such that L|_Z is not big. Note that this result is false in characteristic zero. The following lemma is a fundamental component in the proof of this theorem.
  Let f  X → Y be a finite universal homeomorphism between projective schemes defined over a field k of positive characteristic. Let L be a line bundle on Y. Then L is semiample if and only if f^*L is semiample.

It is easy to show that the lemma holds, as f must factor through a power of Frobenius. 



  Let X be a normal projective variety over a perfect field of characteristic p>0. Assume that A is an ample -divisor and B ≥ 0 is an -divisor such that K_X+B is -Cartier and K_X+A+B is nef and of maximal nef dimension. Then K_X + A + B is big.


This follows from <cit.> after taking a base change to an uncountable algebraically closed field.






 §.§ W-rationality and normality of divisorial centres

In this subsection we recall some definitions and results from <cit.>. First, given a k-scheme X, over a perfect field k of positive characteristic p>0, we define the sheaf W(_X) as follows

    W(_X)(U)  W(_X(U))  for an open subset  U ⊆ X.
 
		Here W Alg_𝔽_p→Alg_ is the Witt vector functor (see <cit.>).

Let (X) be the abelian category of sheaves of abelian groups on X, and let (X)_ be the quotient category

    (X)_(X)   /  {∈(X) | n = 0  for some  n ∈_>0}.

We set W_X, to be the image of W_X in (X)_ under the natural projection. Given a morphism of k-schemes f  Y → X, one can define derived functors R^if_*(W_Y,) for i≥ 0.
 We say that a k-variety X for a perfect field k of positive characteristic has W-rational singularities, if there exists a (equivalently, for any) resolution of singularities ϕ Y → X such that R^iϕ_*(W_Y,) = 0 for i>0.


The following result asserts that normality of divisorial centres and W-rationality are preserved under pl-contractions.
 
Let (X,Δ) be a -factorial three-dimensional dlt pair defined over a perfect field k of characteristic p>0, with normal divisorial centres up to a universal homeomorphism. Let g  X → Z be a projective birational morphism onto a normal variety Z such that

	
  * -(K_X+Δ) is g-ample,
	
  * -S is g-ample for an irreducible component S of ⌊Δ⌋.
	

Then the irreducible components of ⌊ g_*Δ⌋ are normal up to a universal homeomorphism. Moreover, if X has W-rational singularities, then so does Z.


Lastly, let us state the following result which shows that W-rationality and normality of divisorial centres up to a universal homeomorphism are preserved by curve extractions. The proposition will be used to study the behaviour of these notions under flips as in <cit.>.
  Let k be a perfect field of characteristic p>0, and let g  Y → Z be a proper birational morphism between normal threefolds over k such that every fibre of g is at most one-dimensional. Assume that (Y, Δ) is dlt for some effective -divisor Δ. Then

	
  * if R^1g_*(W_Y,) = 0 and irreducible components of ⌊ g_*Δ⌋ are normal up to a universal homeomorphism, then (Y,Δ) has normal divisorial centres up to a universal homeomorphism;
	
  * if Z has W-rational singularities, then so does Y. 
 


This follows from <cit.>, <cit.>, and <cit.>.

Note that if R^1g_*_Y = 0, then R^1g_*(W_Y,) = 0 (<cit.>).




 §.§ Normality of plt centres under MMP

In this subsection we obtain some variants of the results in <cit.> pertaining to the study of normality of plt centres under the Minimal Model Program.

The first result is a variant of Proposition <ref>.
 
Let (X,Δ) be a -factorial three-dimensional dlt pair over a perfect field k of characteristic p>0. Let ϕ X → Z be a projective birational morphism such that

	
  * -(K_X+Δ) is ϕ-ample,
	
  * every irreducible component ⌊Δ⌋ is normal,
	
  * (X,(1-ϵ)Δ) is F-regular over Z for every 0 < ϵ < 1.

Then, every irreducible component of ϕ(⌊Δ⌋) is normal.


Pick an irreducible component S of ⌊Δ⌋ and set S_Z ϕ(S). It is enough to show that R^1ϕ_*_X(-S) = 0. Indeed, under this assumption, we have that 

    _Z ≃ϕ_*_X →ϕ_* _S

 is surjective, and since the image of this map is naturally identifiable with _S_Z, this shows that ϕ_* _S ≃_S_Z. Thus ϕ|_S  S → S_Z is its own Stein factorisation, and so the normality of S implies the normality of S_Z.

To conclude the proof, we note that 

    R^1ϕ_*_X(-S) = R^1ϕ_* _X( K_X + (1-ϵ)(Δ-S) - (K_X+Δ) + ϵ(Δ-S) ) = 0

for sufficiently small ϵ > 0 by the relative version of <cit.>[Jakub: good reference?]. Here we used that (X, (1-ϵ)(Δ-S)) is relatively F-regular, and -(K_X+Δ) + ϵ (Δ-S) is relatively ample.


For the study of normality of plt centres under flips, we also need the following result.
 
Let ϕ X → Y be a birational morphism of normal varieties such that R^1ϕ_* _X = 0 and (ϕ) ≤ 1. Let S be an irreducible divisor on X such that ϕ(S) is normal. Then S is normal.


Set S_Z ϕ(S). First, we show that R^1ϕ(_S) = 0. To this end, consider the following long exact sequence

    R^1ϕ_*_X → R^1ϕ_*_S → R^2ϕ_*_X(-S).

The term on the left is zero by the assumption, and R^2ϕ_*_X(-S)=0 since (ϕ) ≤ 1. Thus R^1ϕ_*_S = 0.


Let f S̃→ S be the normalisation of S. Consider the following short exact sequence.

    0 →_S →  f_*_S̃→→ 0

for some sheaf  on S. Pushing this down onto S_Z, we get

    0 →ϕ_* _S →ϕ_* f_*_S̃→ϕ_* → R^1ϕ_* _S = 0.

Since S_Z is normal, the morphisms ϕ|_S  S → S_Z and ϕ|_S ∘ f S̃→ S_Z are their own Stein factorisations, and hence

    _S_Z≃ϕ_* _S ≃ϕ_* f_*_S̃.

This implies that ϕ_* = 0, and so = 0, which in turn shows that f is an isomorphism.






§ THE PROOF OF THEOREM <REF>

In this section we prove Theorem <ref>. We begin with the following observation.


  Let (Y,Δ) be a -factorial  dlt pair defined over a field k. 
Assume that there exists a commutative diagram of projective birational morphisms



Y dπ rg     Z ld  

X    



such that X is a normal -factorial variety, (π) ⊆⌊Δ⌋, and g is a small birational morphism. Let R be a g-exceptional curve. Then there exists an irreducible effective π-exceptional divisor E' on Y satisfying R · E' > 0.


Write Δ = E + B, where B≥ 0 and  E = ∑ _i=1^kE_i is the union of all irreducible π-exceptional divisors E_i. Assume by contradiction, that the statement of the lemma is false, that is  R · E_i ≤ 0 for every i ≥ 1. Since X is -factorial, there exists an effective π-exceptional π-antiample divisor on Y, and hence R is negative on some irreducible exceptional divisor. Thus, without loss of generality, we may assume R · E_1 < 0, and so R ⊆ E_1.

Set L  g^*H for a very ample divisor H on Z such that  L and E have no common components. Since g_*E_i 0 for i=1,…, k, there exists a  π-relative curve C on E_1, not contained in ⋃_i ≥ 2 E_i, such that L · C > 0. 

Given that X is -factorial, by pushing forward L to X and pulling it back, we obtain

    L ∼_π, ∑_i≥ 1 -b_i E_i

for some b_i ≥ 0. Since L · C >0 and E_i · C ≥ 0 for i ≥ 2, we get E_1 · C < 0 and b_1 > 0. Therefore,

    L · R = -b_1E_1 · R - ∑_i≥ 2 b_iE_i · R ≥ -b_1E_1 · R > 0,
 
which is a contradiction since L· R=0 by our choice of R and L; here we used the ad absurdum assumption that R · E_i ≤ 0 for all i≥ 1.



 
The above lemma holds for X being a normal integral separated -factorial algebraic space of finite type over k (see 083Z). Here, we call an algebraic space ℚ-factorial if every Weil divisor is Cartier (that is Cartier up to a surjective étale cover by a scheme, cf. 083C). The key case we will use in this article is when X admits a projective birational morphism ϕ X' → X such that X' is a normal -factorial variety, ρ(X'/X)=1, and (ϕ) is a divisor. To verify that X is -factorial, we can replace it by an étale cover and assume that it is a variety, in which case the -factoriality follows by a standard argument.
The above lemma holds for X being an algebraic space admitting a projective birational morphism ϕ X' → X such that X' is a normal -factorial variety, ρ(X'/X)=1, and (ϕ) is a divisor. Indeed, since the statement is étale local on X, we can replace X by an étale cover and assume it is a scheme. Now consider the following diagram



W rϕ_W dπ_W     Y dπ rg     Z ld  

X' rϕ     X,    



where W is smooth and π_W, ϕ_W are birational. With notation as in the above proof, we claim that

    L ∼_π, ∑_i≥ 1 -b_i E_i,

where L = g^*H for a general very ample divisor H on Z. This was the most important part of the proof which necessitated X being a -factorial variety. To this end, set L_W ϕ^*_W L and L_X'π_W* L_W. Let D (ϕ) and D_W be the strict transform of D on W. Note that D is an irreducible divisor as ρ(X'/X)=1. Since L = g^*H for a general hyperplane section H, we can assume that L_W and D_W have no common components. In particular, L_X' is non-negative on a general curve in D, hence L_X'∼_,X -aD for some a ∈_≥ 0. Since X' is -factorial, by pushing L_W onto X' and back, we get

    L_W ∼_,X -aD_W + E

where E is a π_W-exceptional -divisor. By the negativity lemma over X', we get that E is anti-effective. Therefore, ϕ_W*(-aD_W + E) ∼_π,  L is anti-effective which concludes the proof of the claim.

By applying the same argument to L replaced by an ample divisor on Y, we get that there exists an ample anti-effective exceptional divisor on Y over X. With this observation and the validity of (<ref>), the proof of Lemma <ref> follows in the above setting without any other change.




 §.§ Flips

In this subsection, we tackle the existence of flips. The following lemma is a key ingredient in the proof of Proposition <ref>.

  Let (S,C + B) be a two-dimensional plt pair defined over an infinite perfect field k of characteristic p>0, where C is an irreducible curve, and let f  S → T be a projective birational morphism onto a surface germ (T,0) such that -(K_S+C+B) and C are f-nef. Then (S,C+B) is relatively purely F-regular over a neighbourhood of f(C) ⊆ T.

 By <cit.> and Stein factorisation, we can assume that T is normal. By the relative base point free theorem, -(K_S+C+B) is semi-ample over T, and hence there exists an effective -divisor D ∼_ -2(K_S+C+B) such that (S,C+B+D) is log canonical (this follows from <cit.> after having taken an lc compactification as in the proof of <cit.>). Replacing B by B+1/2D, we can suppose that -(K_S+C+B) ∼_f,  0 and (S,C+B) is plt. 

Let E  f_*C and B_T  f_*B. Since C is irreducible and f-nef, one sees that E 0 is an irreducible curve. Moreover, we have that K_S+C+B = f^*(K_T + E + B_T), and so (T, E+B_T) is plt. In particular, E is normal, and (E , B_E) is klt where K_E+B_E=(K_T+E+B_T)|_E. Since E=1, (E , B_E) is log smooth and so it is strongly F-regular. Consequently, F-adjunction shows that  (T,E+B_T) is purely F-regular (see for instance Lemma <ref>), and <cit.> implies that K_S+C+B is relatively purely F-regular.



 Now let R be an effective f-exceptional divisor such that -R is ample, and let t>0 be the log canonical threshold of (S,C+B) with respect to R. Let g  Z → S be the dlt modification of (S,C+B+tR) and write K_Z+Δ_Z = g^*(K_S + C + B + tR).

Let R' be an effective g-exceptional divisor such that -R' is g-ample. Take 0 < ϵ≪ 1 and set Δ'_Z Δ_Z - ϵ g^*R + t' R', where t' is the log canonical threshold of (Z, Δ_Z - ϵ g^*R) with respect to R'. In particular, -(K_Z + Δ'_Z) is ample. Furthermore, (Z,Δ'_Z) is dlt given ϵ has be chosen to be sufficiently small.

By construction, there exists an exceptional irreducible curve E ⊆⌊Δ'_Z ⌋. Write K_E + Δ'_E = (K_Z+Δ'_Z)|_E. Since f^-1_*C ⊆Δ'_Z and the non-klt locus of (Z,Δ'_Z) is connected, we get that (E,Δ'_E) is lc but not klt. Hence (E,Δ'_E) is purely-F-regular by F-adjunction (see for instance <cit.>), and so is (Z,Δ'_Z) by 




By [HX, Theorem 3.2], there exists an N-complement (S,B^c) for N≤ 6, which is lc but not klt. Since m>6, we must have that C ⊆⌊ B^c ⌋. Let v S̃→ S be its dlt-modification, with K_S̃ + B^c_S̃ = v^*(K_S + B^c) and C_S̃ = ⌊ B^c_S̃⌋. 

If (S̃, B^c_S̃) is plt, then C_S̃ is exceptional over T (see the proof the [HX, Theorem 3.2] or the last line of page 723). But the strict transform of the f-ample divisor C is contained in B^c_S̃ which leads to a contradiction.

Hence (S̃, B^c_S̃) is not plt and (C_S̃, Diff_C_S̃(B^c_S̃ - C_S̃) is of type (2,2, ∞) or (∞, ∞). In particular, by global F-adjunction, it is F-split.

Following the proof of [HX, Theorem 3.1], we see that it is enough to show that (ℙ^1, a_1P_1 + a_2P_2 + a_3P_3) is globally F-regular, where a_1,a_2 ≤1/2 and a_3 < 1, or a_1,a_2 < 1 and a_3=0. This is true in any characteristic.
 
 

The following proposition in conjunction with Lemma <ref> gives the existence of flips as in Theorem <ref>.
  Let (X,Δ) be a three-dimensional -factorial dlt pair defined over a perfect field k of characteristic p>0. Let ϕ X → Z be a flipping contraction of a (K_X+Δ)-negative extremal ray R. Suppose that there exist irreducible divisors E, E' ⊆⌊Δ⌋ such that R · E < 0 and R ·  E' > 0. Then the flip (X^+, Δ^+) of f exists. 

Moreover, if (X,Δ) has normal divisorial centres up to a universal homeomorphism, then so does (X^+, Δ^+).


 Note that if (X,S + S' + B) is a three-dimensional dlt pair such that S, S' are distinct irreducible divisors and ⌊ B ⌋ = 0, then (S̃, Δ _S̃) is plt, where S̃ is the normalisation of S and  K_S̃ + Δ _S̃ = (K_X +S+S'+B)|_S̃. This can be verified on a log resolution of (X,S+S'+B).



 We work over a neighborhood of the closed point P=ϕ (R)⊂ Z and hence we will frequently replace Z by an appropriate neighborhood of P∈ Z. By base change to its algebraic closure, we can assume that k is algebraically closed. Except for -factoriality, all the assumptions are preserved under this base change (cf. <cit.>). Moreover, it is enough to construct a flip after this base change by <cit.>. Although X may not be -factorial, the irreducible components of Δ are -Cartier and this is all that we need in the following proof.

Write Δ = E + E' + D, where D ≥ 0, and set 

    Δ^plt E + (1-ϵ)(E'+D)

for some ϵ > 0 sufficiently small so that (K_X + Δ^plt) · R < 0  and (K_X + Δ^plt + ϵ E') · R < 0. 

We aim to show that (X, Δ^plt) is relatively purely F-regular over Z. To this end, write 

    K_Ẽ + Δ_Ẽ   = (K_X + Δ^plt)|_Ẽ,  and 
    
    	K_Ẽ + Δ'_Ẽ   = (K_X + Δ^plt + ϵ E')|_Ẽ,

where g Ẽ→ E is the normalisation of E. 
Since (X, Δ) is dlt, the above remark shows that (Ẽ, Δ'_Ẽ) is plt, and so Lemma <ref> implies that it is, in fact, relatively purely F-regular over a neighborhood of P ∈ϕ(E). Hence we may assume that (Ẽ, Δ_Ẽ) is relatively F-regular.

By inversion of F-adjunction (see Lemma <ref>), this implies that (X, Δ^plt) is relatively purely F-regular over Z. In particular, the proof of the existence of the flip of (X, Δ^plt) from <cit.> holds without any change (see Remark <ref>). Since such a flip is also the flip of (X,Δ), we obtain (X^+, Δ^+) sitting inside the following diagram



X [swap]rdϕ [dashed]rrψ         X^+ ldϕ^+ 

    Z,    



where Δ^+ ψ_* Δ. 

In order to show the normality of divisorial centres of ⌊Δ^+ ⌋ up to a universal homeomorphism, we proceed as follows. First, we have that irreducible components of ϕ_*⌊Δ⌋ are normal up to a universal homeomorphism by Proposition <ref>. 

By <cit.>, since X is relatively F-regular, we have that X^+ is relatively F-regular as well. Pick an ample divisor A such that R^1ϕ^+_* _X^+(A) = 0. Since F^e _X^+→ F^e_* _X^+(A) splits locally over Z for some e>0, we get that

    R^1ϕ^+_*_X^+⊆ F^e_* R^1ϕ^+_*_X^+(A) = 0.

In particular, R^1ϕ^+_*(W_X^+, )=0 by <cit.>. By Proposition <ref>, this implies that (X^+,Δ^+) has normal divisorial centres up to a universal homeomorphism.
 


  In the proof of the above theorem, we used the fact that if ϕ X → Z is a (K_X+S+B)-flipping contraction of an extremal curve R for a three-dimensional relatively purely F-regular pair (X,S+B) defined over a perfect field of characteristic p>0 with S being an irreducible divisor such that R · S < 0, then the flip of ϕ exists. This has been implicitly proven in <cit.>. 

The statement of <cit.> assumes that the coefficients of Δ^plt are standard, but this is only used to show that (X, S+B) is relatively purely F-regular, which we assume apriori to be true. Moreover, the results of <cit.> are stated over an algebraically closed field, but it is enough to show the existence of flips after base-changing to the algebraic closure k of k (see <cit.>). Here note that the -factoriality of X need not be preserved, but fortunately -factoriality is not needed in the proof of <cit.>.


For the convenience of the reader, we recall the general strategy of the proof.

First, by <cit.> we have that S is normal. Now, fix an effective Cartier divisor Q ∼ k(K_X+S+B) on X for some k>0 sufficiently divisible such that the support of Q does not contain S, and define the following b-divisors

    𝐍_i 𝐌𝐨𝐛(iQ),   𝐌_i 𝐍_i|_S,   and  𝐃_i 1/i𝐌_i.

Write K_S + B_S = (K_X+S+B)|_S, and let S̅ be the terminalisation of (S,B_S). Then 𝐌_i descends to S̅ by <cit.>.

Following the strategy depicted in <cit.>, it is enough to show that 𝐃_i stabilise for i≫ 0. To this end, we note that the ℝ-divisor 𝐃lim𝐃_i is semiample on S̅ by <cit.>. Let a S̅→ S^+ be the induced fibration as in <cit.>. By <cit.>, we have that 𝐃_S̅ is rational and a_*𝐃_S̅ = a_*𝐃_j, S̅ for divisible enough j≫ 0.

Let g  Y → X be an appropriately chosen resolution of singularities such that 𝐍_i,Y is free. We replace Y upon changing i. Set L_i,j⌈j/i𝐍_i,Y + 𝐀_Y ⌉ for i,j>0 and 𝐀 being the discrepancy b-divisor for (X,S+B). Explicitly:

    K_Y + S' = f^*(K_X + S + B) + 𝐀_Y,

where S' is the strict transform of S. Since (S,B_S) is relatively purely F-regular, the proof of <cit.> holds without any change, and so

    S^0(S', σ(S', ψ_S') ⊗_S'(L_i,j|_S')) = H^0(S^+, _S^+(j𝐃_S^+))

for j replaced by some multiple and i ≫ 0 divisible by j, where the left hand side is the subset of sections in H^0(S', _S'(L_i,j|_S')) which are stable under the Frobenius trace map of (S', ψ_S') (see <cit.> for the precise definition). Here ψ_S' (ψ - S')|_S' and ψ is a small perturbation of {-j/i𝐍_i,Y - 𝐀_Y}+S' (see <cit.>). Moreover, the natural morphism

    S^0(Y, σ(Y,ψ)⊗_Y(L_i,j)) →     S^0(S', σ(S', ψ_S') ⊗_S'(L_i,j|_S')) 
       = H^0(S^+, _S^+(j𝐃_S^+))

is surjective (see <cit.>). Since

    |L_i,j| ⊆ |jk(K_Y + S' + B_Y)| + ⌈ A_Y ⌉,

where B_Y = (-A_Y)_≥ 0, this allows for lifting sections and showing the stabilisation of 𝐃_i (see the end of the proof of <cit.> for details).  

To sum up, the relative pure F-regularity of (X,S+B) is used twice: to show the normality of S, and, more importantly, to establish the identity (<ref>) which is fundamental in lifting sections. 




 §.§ Proof of Theorem <ref> and Theorem <ref>



For inductive reasons and the sake of clarity, we tackle Theorem <ref>, Theorem <ref>, and  Corollary <ref> simultaneously. Using the following proposition we will show that every  three-dimensional dlt pair has normal divisorial centres up to a universal homeomorphism, which in turn will conclude the proof of Theorem <ref>. This approach follows, mutatis mutandis, the strategy laid down in <cit.>.
  Let (Y,Δ) be a -factorial three-dimensional dlt pair defined over a perfect field of characteristic p>0 with normal divisorial centres up to a universal homeomorphism. Assume that there exists a projective morphism π Y → X over a normal -factorial variety X such that (π) ⊆⌊Δ⌋. 

Then we can run a (K_Y+Δ)-MMP over X which terminates with a minimal model (Z,Δ_Z), whose divisorial centres are normal up to a universal homeomorphism. Moreover, if Y has W-rational singularities, then so does Z.


The cone theorem is valid by Theorem <ref>.  Since X is -factorial, there exists a relatively anti-ample effective exceptional divisor. In particular, there exists S ⊆⌊Δ⌋ such that S · R < 0 for a given (K_X+Δ)-extremal ray R. Proposition <ref> implies the existence of the contraction g  X → Z of R. By Proposition <ref>, the normality of divisorial centres up to a universal homeomorphism is preserved. By Lemma <ref> and Proposition <ref>, if g is a flipping contraction, then the flip exists and the normality of divisorial centres up to a universal homeomorphism is preserved under g. The termination of the MMP with scaling holds by standard arguments (cf. <cit.> and <cit.>). Thus, a minimal model exists.

Each step of the MMP preserves the W-rationality of singularities by Proposition <ref> and Proposition <ref>. Therefore, if Y has W-rational singularities, then so does Z.


In order, to show Theorem <ref>, we need to prove Theorem <ref> first.

Let π Y → X be a log resolution of (X, S+B). By Proposition <ref>, we can run a (K_Y+π^-1_*(S+B) + (π))-MMP over X. The corresponding minimal model must be (X,S+B) itself. In fact the plt condition guarantees that all the components of (π) are contracted and hence the minimal model is isomorphic to X in codimension one. Since X is -factorial, the above minimal model is in fact isomorphic to X. By Proposition <ref>, the normalisation of S is a universal homeomorphism. 



By Theorem <ref> we have that the divisorial centres of (Y,Δ) are normal up to a universal homeomorphism. The theorem now follows from Proposition <ref>.


Corollaries <ref>, <ref> and <ref> now follow.

Let π Y → X be a log resolution of X. By Proposition <ref>, we can run a (K_Y+(π))-MMP over X, the minimal model of which must be X itself, as it is klt and -factorial. In particular, X has W-rational singularities by Proposition <ref>.



Let π Y → X be a log resolution of (X, Δ). Then, a dlt modification is a minimal model of (Y,π^-1_*Δ + (π)) over X (see Theorem <ref>).



By taking a log resolution of (X,S+B) it is easy to see that if (X,S+B) is plt, then (S̃,B_S̃) is klt. Thus, we can assume that (S̃,B_S̃) is klt and aim to show that (X,S+B) is plt near S. 

Let π Y → X be a dlt modification of (X,S+B) (see Corollary <ref>) and write K_Y + S_Y + B_Y = π^*(K_X+S+B). By definition (of a dlt modification) for any π-exceptional irreducible divisor E we have that E ⊆⌊ B_Y ⌋.  Write

    (π|_S_Y)^*(K_S̃ + B_S̃) = (K_Y+S_Y+B_Y)|_S̃_Y = K_S̃_Y + B_S̃_Y,

where S̃_Y → S_Y is the normalisation of S_Y, and B_S̃_Y is the different. Let E be a π-exceptional divisor intersecting S_Y. Since E ⊆⌊ B_Y ⌋ and (Y,S_Y+(π)) is dlt, we must have that E ∩ S_Y ⊆⌊ B_S̃_Y⌋. If E ∩ S_Y ≠∅, then this contradicts (S̃, B_S̃) being klt. 

Therefore we may assume that E∩ S_Y=∅ so that Y=X near S and hence (X,S+B) is dlt on a neighborhood of S. Since S is irreducible, (X,S+B) is in fact plt.



  Assume that Y is projective. Then Theorem <ref> holds in a slightly more general setting: suppose that X, instead of being a normal -factorial variety, is an algebraic space admitting a birational morphism ϕ X' → X such that X' is a normal -factorial variety, ρ(X'/X)=1, and (ϕ) is a divisor. In view of Remark <ref>, the assertions of Theorem <ref> hold true by exactly the same proof. Here, the assumption on the projectivity of Y is needed in order to apply the cone theorem.




  In fact all log canonical centres of a three-dimensional -factorial dlt pair (X,Δ) are normal up to a universal homeomorphism. Indeed, if the centre is two-dimensional, then this follows from Theorem <ref>. Thus, we can assume that we have a dlt pair (X,D_1+D_2) and we need to show that every irreducible component of D_1 ∩ D_2, say C, is normal up to a universal homeomorphism. Let u  D^n_1 → D_1 be the normalisation of D_1 and let C'  u^-1(C). By adjunction (cf. Corollary <ref>), (D_1^n, Diff) is plt, where K_D^n_1 + Diff = (K_X+D_1+D_2)|_D^n_1. By the surface theory, C' ⊆⌊Diff⌋ is smooth. In particular, the universal homeomorphism u|_C' C' → C is also the normalisation. 




§ MINIMAL MODEL PROGRAM FOR DLT REDUCTIONS

In this section we show Theorem <ref>. The main ingredient of the proof is the following result.
  Let (X,S+B) be a three-dimensional -factorial plt pair defined over a perfect field k of characteristic p>0, where S is an irreducible divisor. Let g  X → Z be a flipping contraction of a K_X+S+B negative extremal curve R such that R ⊆ S and  R · S = 0. Then the flip of g exists.

The proof follows the same strategy as the reduction of the existence of flips to pl-flips (<cit.>). The necessary pl-flips exist by Proposition <ref>. 
 It suffices to prove the claim in a neighborhood of the point P :=g(R)⊂ Z. Since S · R = 0, we have that S'  g(S) is -Cartier. Set Δ = S+B and let H' be a reduced Cartier divisor on Z  containing the  non simple normal crossing locus of (Z, g_*Δ) and satisfying the following conditions (see assumptions (i)-(iv) in <cit.>):

	
  * H  g_*^-1H' contains (g),
	
  * for any proper birational morphism h  Y → Z such that Y is -factorial, we have that N^1(Y/Z) is generated by the irreducible components of the strict transform of H' and the h-exceptional divisors,
	
  * H and Δ have no irreducible components in common.

Consider a log resolution of (X,Δ + H):



Y rp [bend left = 35]rrh     X rg     Z.




We claim that we can run a (K_Y+Δ_Y+H_Y)-MMP over Z, where H_Y is the strict transform of H, and Δ_Y  p_*^-1Δ + (h). 
First of all note that for any extremal ray R for Y over Z, we may assume that R is contained in the support of h^*H' and by condition (2) there is a component of the support of h^*H' having a non-zero intersection number with R. Since R· h^*H'=0, there are components  E,E' of  the support of h^*H' such that E· R<0 and E'· R>0. 
Divisorial centres are normal up to a universal homeomorphism by Theorem <ref>, and so contractions exist by Proposition <ref>. The cone theorem holds by Theorem <ref>, and  special termination is valid by the standard argument (see <cit.>). 
The necessary flips exist by Proposition <ref>.




Let us replace (Y,Δ_Y+H_Y) by its minimal model over Z. Note that, after this replacement, Y need not admit a map to X, but it still admits a map to Z, which as above we denote by h  Y → Z. Write Δ_Y = D + B_Y, where B_Y=h^-1_*g_* B so that ⌊ B_Y ⌋ = 0 and D = ∑_i=1^m D_i is a sum of irreducible divisors not contained in H_Y with D_1 = S_Y being the strict transform of S and D_2,… , D_m are the h-exceptional divisors. Since h^*H' ≡_h 0, we have that

    H_Y ≡_h -∑_j b_j D_j,  where  b_j ∈_≥ 0.

Run a (K_Y+Δ_Y)-MMP with scaling of H_Y. Let 0 < λ≤ 1 be such that K_Y+Δ_Y + λ H_Y is h-nef, and there exists a (K_Y+Δ_Y)-extremal curve R satisfying (K_Y+Δ_Y+λ H_Y) · R = 0. Since (K_Y+Δ_Y) · R < 0, we have that H_Y · R > 0, and the above numerical equivalence implies that there exists j for which D_j · R < 0.  Arguing as above, in order to show that such an MMP can be run, it is enough to show that flips exist. Hence we suppose that the associated contraction is flipping, and so in particular h(R)=P. We have h^*S' · R = 0 and g(R)=P, and thus there exists j' for which D_j'· R ≥ 0 and D_j'∩ R ≠∅. We claim that D_j'· R > 0. Indeed, otherwise R ⊆ D_j', and since (Y, D_j + D_j') is dlt, we get that R = D_j ∩ D_j'. Then

    R · D_j' = R|_D_j· D_j'|_D_j = λ (R|_D_j)^2 < 0,

for some λ >0, which is a contradiction. Given that D_j · R < 0 and D_j'· R > 0, the flip of R exists by Proposition <ref>.

Now, replace (Y,Δ_Y) by the output of this MMP. Then K_Y+Δ _Y is nef over X.  Notice that h is small (by the negativity lemma since (X,S+B) was plt) and (Y,Δ_Y) is the flip of (X,Δ). It is then easy to see that  ρ (Y/X)=1 (cf. <cit.>) and so  K_Y+Δ _Y  is in fact ample over X. 
  


Note that X is not of finite type over k and hence is not a variety. In what follows, when we invoke results about varieties, we implicitly refer to the base change of (𝒳, Φ) with respect to an appropriate open neighbourhood of s ∈ C.

We run a (K_X+Δ)-MMP with scaling of A, a sufficiently ample divisor over S  Spec(R). We need to show that the cone theorem and the base point free theorem are valid for (X,Δ), that the flips exist, and that the MMP terminates. Let A be an ample -Cartier -divisor on X.


Cone theorem Let X_s = ⋃_j=1^m E_j be the reduced special fibre of π, where E_j are irreducible components, and let Ẽ_j → E_j be the normalisations. Take {Γ_j,k}_k ≥ 0 to be the  finite collection of images of  

    K_Ẽ_j + Δ_Ẽ_j + A|_Ẽ_j = (K_X+Δ+A)|_Ẽ_j

negative extremal curves in Ẽ_j. Since R is a DVR, NE(X/S) is generated by curves contained in X_s, and so       

    NE(X/S) = NE(X/S)_K_X+Δ + A ≥ 0 + ∑_j,k[Γ_j,k].




Base point free theorem
Suppose that K_X+Δ+A is nef and (X,Δ+A) is dlt. We aim to show that it is semiample. By <cit.>), it is enough to check that (K_X+Δ+A)|_X_s and (K_X+Δ+A)|_X_η are semiample where X_s and X_η are the reduced special fibre, and the generic fibre, respectively. The latter -divisor is semiample by the base point free theorem for surfaces (see <cit.>). Thus, it is enough to show that (K_X+Δ+A)|_X_s is semiample.

Let π W → X_s be an S_2-fication (cf. <cit.>) of X_s. By Remark <ref>, all the log canonical centres of (X,X_s) are normal up to a universal homeomorphism, and so  the proof of <cit.> holds in our setting, showing that π is a universal homeomorphism and W is demi-normal. Therefore, (W,Δ_W) is slc, where K_W + Δ_W = π^*(K_X+Δ)|_X_s. 

By abundance for slc surfaces (see <cit.>), K_W+Δ_W is semiample, and, since π is a universal homeomorphism, (K_X+Δ)|_X_s is semiample as well. 


Existence of flips 
Let g  X → Z be a pl-flipping contraction of a (K_X+Δ)-extremal curve C. In particular, C is contained in the special fibre of ϕ X → S, that is there exists an irreducible divisor  E ⊆ϕ^-1(s) such that C ⊆ E. 

If C · E ≠ 0, then, given C ·ϕ^-1(s) = 0, we get that there exists an irreducible divisor E' ⊆ϕ^-1(s) satisfying sgn(C · E') = -sgn(C · E). Thus the flip of g exists by Proposition <ref>. If C · E = 0, then the flip exists by Proposition <ref>.




Termination All the flipped curves are contained in ϕ^-1(s) ⊆⌊Δ⌋, and so the MMP terminates by special termination (see <cit.>).



By the same argument as in the base point free part of the above theorem, we cna show the following result.

  Let k be a perfect field of characteristic p>0 and let (X,Δ) be a ℚ-factorial three-dimensional log canonical pair over  an affine k-variety Z. Suppose that L := K_X+Δ is relatively nef and big, 𝔼(L) ⊆⌊Δ⌋, and 𝔼(L) is projective over Z. Then K_X+Δ is semiample.[CH: Maybe we can remove this lemma for now....since it is giving us a little trouble.... Jakub: I modified it, so that it is exactly what is needed in the next paper.]

Here 𝔼(L) is a union of subvarieties V ⊆ X over Z such that L|_V is not relatively big.

Let π Y → X be a dlt modification of (X,Δ) and write L_Y := K_Y + Δ_Y = π^*(K_X+Δ). We have that 𝔼(L_Y) ⊆ W := ⌊Δ_Y ⌋. By <cit.> and <cit.>, it is enough to show that L_Y|_W is semiample.

Let ϕ W' → W be an S_2-fication (cf. <cit.>) of W. By <cit.>, all the log canonical centres of (Y,Δ_Y) are normal up to a universal homeomorphism, and so the proof of <cit.> holds in our setting, showing that ϕ is a universal homeomorphism and W' is demi-normal. Therefore, (W',Δ_W') is slc, where K_W' + Δ_W' = ϕ^*(K_Y+Δ_Y)|_W. 

By abundance for slc surfaces (see <cit.>[Jakub: Hiromu assumes projectivity over k. Is there a clean way to compactify an slc surface?]), K_W'+Δ_W' is semiample, and, since ϕ is a universal homeomorphism, (K_Y+Δ_Y)|_W is semiample as well. 

The assumption that 𝔼(L) is projective over Z is needed in the above proof, because <cit.> assumes projectivity. We expect that the result is valid without this assumption. 





§ RATIONAL CHAIN CONNECTEDNESS OF LOG FANO VARIETIES

The goal of this section is to show Theorem <ref>. To this end we need to establish a variant of the canonical bundle formula from <cit.>.



 §.§ A variant of a canonical bundle formula

Throughout this subsection we extensively use the following definition.
 Let X be a scheme. We say that an open subset U ⊆ X is big if X  \  U has codimension at least two in X.

We start by showing the following lemma.
 
Let (X,Δ) be a  quasi-projective three-dimensional log canonical pair defined over a perfect field of characteristic p>0. Let ϕ X → Z be a projective morphism to a surface Z such that (K_X+Δ)|_X_η is nef, where X_η is the generic fibre. Then there exists a big open subset U ⊆ Z such that if V ϕ^-1(U), then
(V, Δ|_V) admits a minimal model over U.


Since (K_X+Δ)|_X_η is nef, by <cit.>, it is semiample, and so K_X+Δ is semiample over an open subset of Z. Let R ⊆ Z be the locus of points over which K_X+Δ is not relatively semi-ample.  Note that since the base locus of of m(K_X+Δ) over Z is closed for every m≥ 0, it follows that 𝐁(K_X+Δ /Z) is closed. Since ϕ is a projective morphism, R is also closed.  If R = 0, then we can take U  Z  \  R, concluding the proof. Thus, we can assume that there exists a curve C ⊆ R such that K_X+Δ is not semiample over a neighborhood of the generic point ξ_C of C. 

Localising at ξ_C and running a relative two-dimensional MMP, we can assume that there exist
 
	
  * an open subset U ⊆ Z containing ξ_C,
	
  * a log canonical pair (W,Δ_W),
	
  * morphisms ϕ V  W  U for V ϕ^-1(U),

such that ψ_W is birational, Δ_W  (ψ_W)_*Δ, K_W+Δ_W is nef over ξ_C.  By  <cit.>), after shrinking U, we may assume that  K_W+Δ_W is relatively semiample and

    W  \ ϕ^-1_W(C) ≃ V  \ ϕ^-1(C).

Therefore, by means of gluing, we can construct a variety W together with locally projective morphisms VWU such that

	
  * U U ∪ (Z \  C), and Vϕ^-1(U),
	
  * W is glued from W and X \ ϕ^-1(C) along W \ ϕ^-1_W(C).
 
We may assume that U⊆ Z is a big open subset, and that ψ_W and ϕ_W are projective. Indeed, we can replace U by a big open subset so that ψ_W decomposes into a sequence of steps of a (K_V + Δ|_V)-MMP over U. This, in turn, implies that ψ_W decomposes into a sequence of steps of a (K_V + Δ|_V)-MMP over U, and each such step preserves the projectivity.

By iteratively running the above procedure for (X,Δ) replaced by 

    (W, (ψ_W)_*Δ|_V)

and Z replaced by U, we can conclude the proof.


  Let (X,Δ) be a three-dimensional normal quasi-projective sub-klt pair defined over an algebraically closed field of characteristic p>2 and satisfying κ(-Δ_<0)=0, where Δ_<0 is the negative part of Δ. Let ϕ X → Z be a projective equidimensional morphism to a normal surface Z such that Δ_<0 is vertical, and K_X+Δ∼_ϕ^*L for a -Cartier -divisor L on Z.

Then there exists a purely-inseparable morphism f  T → Z, a rational number 0 ≤ t ≤ 1, and a -divisor Δ_T on T such that κ(-(Δ_T)_<0) = 0 and

    f^*L ∼_ tK_T + (1-t)f^*K_Z + Δ_T.
   
Moreover, if Δ_>0 is big, then (Δ_T)_>0 is big.


By replacing (X,Δ) by its log resolution, we can assume that (X,Δ) is log smooth. Set G  -Δ_<0 and Δ^+ Δ_>0. In particular, K_X+Δ^+ ∼_ϕ,  G and (X, Δ^+) is klt. Note that ϕ need no longer be equidimensional, but we can assume that κ(G|_V)=0, where V ϕ^-1(U) and U is any big open subset of Z.

Let (W,Δ^+_W) be a minimal model of (V,Δ^+|_V) over some big open subset U ⊆ Z, where V ϕ^-1(U) is given by Lemma <ref>. Replacing U by a big open subset, we can assume that ϕ|_V is equidimensional. Consider the following factorisation



V rψ_W [bend left = 35]rrϕ     W rϕ_W     U,



and set G_W  (ψ_W)_*(G|_V), Δ^+_W  (ψ_W)_*(Δ^+|_V). By construction, 

    G_W ∼_ϕ_W,  K_W + Δ^+_W

is relatively nef over U. Moreover, G_W is effective, vertical, and it satisfies κ(G_W)=κ(G|_V)=0. Thus G_W = (ϕ_W)^*(G_Z|_U) for some effective -divisor G_Z on Z satisfying κ(G_Z)=0 (see e.g. <cit.>). In particular, K_W + Δ^+_W ∼_ (ϕ_W)^*(L+G_Z). 

Now, <cit.> applied to (W, Δ^+_W) and ϕ_W  W → U implies that there exists a purely inseparable morphism f  T → U, a rational number 0≤ t ≤ 1, and an effective -divisor Δ^+_T such that  

    f^*(L+G_Z) ∼_ tK_T + (1-t)f^*K_U + Δ^+_T.

Further, Δ^+_T is big when  Δ^+ is big (note that in order to satisfy the assumptions of <cit.>, as (X,Δ^+) is klt, we can assume that Δ^+ = A + E for an ample -divisor A and an effective -divisor E).  Set Δ_T Δ^+_T - f^*G_Z. By taking a compactification T ⊆T such that f extends to fT→ Z and replacing f by the finite part of the Stein factorisation we can conclude the proof; indeed T ⊆T is then a big open subset, and so the -linear equivalences from above are preserved upon such compactification.









 §.§ Proof of Theorem <ref> and its consequences

First, we need the following lemma.



First, we replace X by a spectrum of a local ring. Now let π Y → X be a resolution of singularities of X. Since X is -factorial, for every divisor D on Y we can find a π-exceptional -divisor E such that D + E ∼_X,  0. Indeed, we can take E := π^*π_*D - D. In particular, ^0(Y) ⊗ = 0 by the negativity lemma.

Since (Y_k') = (Y)_k', we get that ^0(Y_k') ⊗ = 0. Moreover, since (Y_k')/^0(Y_k') is reduced and zero-dimensional, every line bundle on Y_k' descends to Y up to an element of ^0(Y_k'), and so some multiple of it descends to Y on the nose.

We need to show that any divisor M on X_k' is -Cartier. Let M_Y be the strict transform of M on Y_k'. By the previous paragraph, there exists a divisor D on Y such that M_Y ∼_ D_k'. By the first paragraph, we can construct an exceptional divisor E on Y such that D + E ∼_X,  0. Hence, M_Y + E_k'∼_X_k',  0, and so M is -Cartier.    




  By base changing to an uncountable algebraically closed field, we can assume that k is uncountable algebraically closed (see <cit.>)  and X is still -factorial (see Lemma 5.4). 
Let ϕ X  Z be the MRCC-fibration; recall that ϕ is a proper morphism over the generic point of Z. By applying bend-and-break to a resolution of X, one sees that X is uniruled and so Z ≤ 2. If Z = 0, then X is rationally chain connected, thus we can assume that Z = 1 or Z = 2. 


Case 1. Z = 1.


 Since ϕ is proper over the generic point of Z and Z = 1, it must extend to a proper morphism ϕ X → Z. Since ρ(X) > ρ(X/Z), we can pick an extremal ray R in NE(X) which is not contracted by ϕ and  admits a supporting hyperplane, that is a nef -Cartier -divisor N which is numerically trivial only on R. Then, for the ample -divisor 
 
    H  -(K_X+Δ)+N,

 we have that K_X + Δ + H = N.

If n(K_X+Δ + H)=3, then K_X+Δ+H is big by Theorem <ref>, and so K_X+Δ + H' is -effective and negative on R for some -divisor H' ≤ H. In particular, the cone theorem (Theorem <ref>) shows that R is represented by a rational curve. Since each fibre of ϕ is rationally chain connected and R is horizontal, X is rationally chain connected.

 If 1 ≤ n(K_X+Δ+H) ≤ 2, then the nef reduction map ψ X  W of K_X+Δ+H contracts R (cf. <cit.>).  Let η∈ W be the generic point. By construction, X_η is geometrically connected, -factorial, and of Picard rank one (see <cit.>). Hence -K_X_η is ample. Thus, by <cit.>, X_η is rationally chain connected, and so a general fibre of ψ is rationally chain connected (see <cit.>, cf. <cit.>). In particular, there exists a rational curve C in a fibre of ψ which is ϕ-horizontal, and X is rationally chain connected.


Case 2. Z = 2.


Let C ⊆ Z be a curve. Then X|_ϕ^-1(C) admits a section by Tsen's theorem; indeed, by taking a reduction and a resolution of singularities one gets a rational map from a ruled surface over C to X|_ϕ^-1(C). Therefore, Z cannot be uniruled as this would contradict <cit.>.


Take H to be a general -hyperplane section in |-(K_X+Δ)|_ and consider the following commutative diagram which exists by flatification (see <cit.>, c.f. <cit.>)



 Y dϕ_Y rg     X [dashed]dϕ 

 Z' rh     Z,



where Y is normal, h and g are projective birational, and ϕ_Y is flat and projective. Let Δ_Y be such that K_Y + Δ_Y = g^*(K_X+Δ+H). Note that K_Y+Δ_Y ∼_ 0 and κ(-(Δ_Y)_<0) = 0. By perturbing H, we can assume that it is the image of a general very ample -divisor on Y, and so  (Δ_Y)_>0 is big.

By applying Proposition <ref> to ϕ_Y, we get the existence of a purely inseparable morphism f  T → Z', a -divisor Δ_T with (Δ_T)_>0 big and κ(-(Δ_T)_<0)=0, and a rational number 0 ≤ t ≤ 1 such that 

    tK_T + (1-t)f^*K_Z' + Δ_T ∼_ 0.

Since Z is not uniruled, Z' and T are not uniruled either. In particular, by classification of surfaces, κ(K_T) ≥ 0 and κ(K_Z')≥ 0. This implies that 

    κ(-(Δ_T)_<0) = κ(tK_T + (1-t)f^*K_Z' + (Δ_T)_>0) > 0

which is a contradiction.

In fact, in the above theorem it is enough to assume that the base change of X to an uncountable algebraically closed field admits a -factorialisation.



We are now ready to show Corollary <ref>.

The argument is exactly the same as in <cit.> given the validity of Corollary <ref> and Theorem <ref>. The proof is based on the Lefschetz trace formula for cristalline cohomology (<cit.>).

 As part of the proof, we obtain that H^i(X,W_X,)=0 for i>0, where X is a -factorial threefold of Fano type defined over a perfect field of characteristic p>2 (cf. <cit.>).



§ OTHER APPLICATIONS

In this section, we gather some other results on three-dimensional birational geometry  that can be extended to low characteristic.
  Let (X,Δ) be a projective -factorial three-dimensional dlt pair defined over a perfect field k of positive characteristic p>0. Let R be a (K_X+Δ)-negative extremal ray. Assume that the closure of the locus of curves which are numerically equivalent to a multiple of R is two-dimensional. Then the contraction of R exists.



Given Remark <ref>, the proof is exactly the same as that of <cit.>.


We can also generalise the main results of <cit.>.

  Let (X,x) be a three-dimensional -factorial klt singularity defined over a perfect field of characteristic p>0. Then the tame fundamental group π^tame_1(X,x) is finite.


It follows by exactly the same proof as that of <cit.>. The reason for that proof to require the assumption p>5, was the use of the existence of a Kollár component, which now follows from Theorem <ref> and the standard argument.   



 Consequently, we show that the non-vanishing conjecture holds for non-irregular terminal threefolds in characteristic p=5.
  Let X be a terminal -factorial projective minimal threefold defined over an algebraically closed field k of characteristic p>3. Suppose that ^0(X')= 0 for every quasi-étale Galois cover X' → X which is tamely ramified over X^sm[Jakub: shall we keep this proposition with this weird assumption, or just move this to a remark?]. Then κ(K_X)≥ 0.



Again, the proof is exactly the same as that in <cit.>, wherein the assumption on the characteristic was used twice: to get the finiteness of the local tame fundamental group (Propostion <ref>) and the semistability of a tensor product of two semistable sheaves of rank two. The latter requires p>3. [Jakub: shall we mention the comment by Lei that in fact p>3 can be dropped, too?]



As informed by Chenyang Xu and Lei Zhang, one should be able to prove the non-vanishing in characteristic p=5 when ^0(X) ≠ 0 as well (generalising <cit.>), with help of our results on the Minimal Model Program and <cit.>.
[I suggest removing everything in red .... starting from consequently....]


  Let (X,Δ) be a -factorial three-dimensional log canonical pair defined over a perfect field of characteristic p>0 and such that X has klt singularities. Then minimal log canonical centres of (X,Δ) are normal up to a universal homeomorphism.


Let W be a minimal log canonical centre of (X,Δ). By the tie-breaking trick, up to perturbing Δ, we can assume that W is a unique log canonical centre with a unique divisor over X of discrepancy -1 (cf. the proof of <cit.>). If _X(W) = 1, then (X,Δ) is plt, and the proposition follows by Theorem <ref>. Thus, we can assume that _X(W)=2.

Let f  Y → X be a dlt modification of (X,Δ). By construction, f extracts exactly one divisor S and we have that 

    K_Y + S + Δ' = f^*(K_X+Δ),

where Δ'  f^-1_*Δ. Since (Y,S+Δ') is plt, S is normal up to a universal homeomorphism, and so there exists an integer e>0 and a factorisation 

    _S →_S^n→ F^e_*_S,

 where S^n is the normalisation of S. The proof of <cit.> does not require any assumptions on the characteristic (it only needs S to be normal in codimension one which follows by means of localisation and surface theory), hence R^1f_*_Y(-S)=0 and the sequence

    0 → f_*_Y(-S) →_X → f_*_S → 0

is exact. Note that f_*_Y(-S) = _W, where _W is the ideal sheaf of W. 

The surjective morphism on the right factors as _X →_W → f_*_S, which implies that _W ≃ f_*_S. Thus, we have a sequence of injective morphisms

    _W →_W^n→ f_*_S^n→ f_*F^e_*_S ≃ F^e_* _W.

In particular, the normalisation of W factors through the Frobenius F^e, and so it is a universal homeomorphism.

One can also show that <cit.> holds for p>0. The only issue is that the techniques of our paper do  not allow for extracting arbitrary divisors with discrepancies smaller than zero. However, by using the tie-breaking trick as above, the requisite divisors in the proof of <cit.> can be extracted with the help of dlt modifications (see Corollary <ref>).



§ ACKNOWLEDGEMENTS

We would like to thank Fabio Bernasconi, Paolo Cascini, Hiromu Tanaka, Joe Waldron, Chenyang Xu, and Lei Zhang for comments and helpful suggestions.

The first author was partially supported by NSF research grants no: DMS-1300750, DMS-1840190, DMS-1801851 and by a grant
from the Simons Foundation; Award Number: 256202. He would also like
to thank the Mathematics Department and the Research Institute for Mathematical Sciences,
located Kyoto University and the Mathematical Sciences Research Institute in Berkeley were some of this research was conducted. The second author was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1] during his PhD at Imperial College London, by the National Science Foundation under Grant No. DMS-1638352 at the Institute for Advanced Study in Princeton, and by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester.

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