Toricity in families of Fano varieties
Lena Ji Joaquín Moraga


Toricity in families of Fano varieties]
Toricity in families of Fano varieties

L. Ji]Lena Ji
Department of Mathematics, University of Illinois Urbana-Champaign, 273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801
lenaji.math@gmail.com

J. Moraga]Joaquín Moraga
UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA

jmoraga@math.ucla.edu

§ ABSTRACT  
Rationality is not a constructible property in families. In this article, we consider stronger notions of rationality and study their behavior in families of Fano varieties. We first show that being toric is a constructible property in families of Fano varieties.
The second main result of this article concerns an intermediate notion that lies between toric and rational varieties, namely cluster type varieties. 
A cluster type ℚ-factorial Fano variety contains an open dense algebraic torus, but the variety does not need to be endowed with a torus action.
We prove that, in families of ℚ-factorial terminal Fano varieties, being of cluster type is a constructible condition. As a consequence, we show that there are finitely many smooth families parametrizing n-dimensional smooth cluster type Fano varieties.


[
    [
    September 9, 2024
=====================








§ INTRODUCTION
 

Let X → T be a family of projective varieties over ℂ. The subset

    T_ rational{closed  t ∈ T |𝒳_t  is a rational variety, i.e., 𝒳_t is birational to projective space}

is (the set of closed points of) a countable union of locally closed subsets of T
(see <cit.>).
In smooth families, T_ rational is a countable union of closed subsets <cit.>; however, for families with singular members, these subsets are in general neither open nor closed <cit.>. Furthermore,
T_ rational⊆ T is not in general a constructible subset,
i.e., it is not a finite union of locally closed subsets (in the Zariski topology) <cit.>.
There are also families where T_ rational = T, but the generic fiber is irrational as a k(T)-variety (see, e.g., <cit.>).

From the perspective of the Minimal Model Program (MMP) it is natural to investigate the behavior of rationality in families of Fano varieties.
In the case of singular degenerations of Fano varieties, Birkar, Loginov, and Qu have shown that the dimension and log discrepancies of general fibers control the irrationality of the special fiber <cit.>.

One important class of rational varieties is the class of toric varieties, i.e., those that contain a dense algebraic torus whose action on itself extends to the whole variety.
Our first result is that being toric is a constructible condition in families of (klt) Fano varieties:


Let 𝒳→ T be a family of Fano varieties over ℂ.
Then, the set 

    T_ toric{closed  t ∈ T |𝒳_t  is a toric variety}

is a constructible subset of T. 

It is expected that the previous theorem should hold without the Fano assumption.
We note that being toric is neither an open nor a closed condition (Examples <ref> and <ref>).

Being toric places strong restrictions on the geometry of a variety.
For instance, smooth toric Fano varieties do not deform <cit.> (see also <cit.>), so Theorem <ref> is automatic in the smooth case.
Thus, we consider an intermediate class: cluster type varieties (see Definition <ref>). This is a class of rational varieties generalizing toric varieties, and was introduced by Enwright, Figueroa, and the second author in <cit.>.
For example, smooth toric varieties are covered by tori, whereas smooth cluster type Fano varieties contain an open dense torus (see <cit.>).
We prove the following result about cluster type varieties in families of Fano varieties:


Let 𝒳→ T be a family of ℚ-factorial terminal Fano varieties over ℂ.
The set 

    T_ cluster{closed  t∈ T |𝒳_t  is a cluster type variety}

is a constructible subset of T. 


The -factorial terminal assumption in the previous theorem is vital for our proof to work (see Remark <ref>).
Many properties of families of terminal Fano varieties do not hold if we drop the condition
on the singularities (see, e.g., <cit.>).
As a consequence of Theorem <ref> and a result of Kollár–Miyaoka–Mori <cit.>, we get the following corollary about cluster type smooth Fano varieties.

Let n be a positive integer. 
Then, there are finitely many smooth projective morphisms f_i𝒳_i→ T_i parametrizing 
n-dimensional smooth cluster type Fano varieties over ℂ.

By parametrizing, we mean here that every n-dimensional smooth cluster type Fano appears as the closed fiber of some f_i, and, moreover, every closed fiber of each f_i is in this class of varieties.
This notion is stronger than boundedness,
but weaker than moduli, since the same variety could appear as a fiber multiple times.

Furthermore, from the proof of Theorem <ref>, it follows that all fibers of each f_i in Corollary <ref> admit the same combinatorial structure, in the sense that a certain birational transformation to a toric pair can be performed in the whole family (see Lemma <ref>).

For smooth rational Fano varieties, it is not clear whether the analogous statement to Corollary <ref> should be true. In the toric case, the analogous statement was already known: smooth toric Fano varieties correspond to smooth reflexive polytopes (see, e.g., <cit.>), so in each dimension n there are only finitely many up to isomorphism.
Note that smooth cluster type Fanos do deform, so unlike in the toric case,
the parametrizing varieties T_i in Corollary <ref> are in general positive-dimensional.
For example, every smooth del Pezzo surface of degree ≥ 2 is of cluster type by <cit.> and <cit.>.



 §.§ Outline

In Section <ref>, we begin by recalling definitions and preliminary results on singularities of pairs, toric log Calabi–Yau pairs, and cluster type log Calabi–Yau pairs. We prove several lemmas that we will apply in later sections.
Next, in Sections <ref> and <ref>, we prove versions of Theorems <ref> and <ref> for families of Fano varieties with a log Calabi–Yau pair structure. Then, in Section <ref>, we use our results from Sections <ref> and <ref> to prove the main theorems. Finally, in Section <ref>, we end the article with examples of different behaviors of toricity in families, and we pose some related questions.



 §.§ Acknowledgements

We thank Stefano Filipazzi, János Kollár, and Burt Totaro for helpful conversations and comments.
Part of this work was carried out while the authors were visiting the Yau Mathematical Sciences Center at Tsinghua University, and we thank Caucher Birkar and Spring Ma for their hospitality.



§ PRELIMINARIES


We work over an algebraically closed field of characteristic 0. 
We write Σ^n for the sum of the coordinate hyperplanes of ^n. Throughout, a Fano variety will mean a variety X such that -K_X is ample and X has klt singularities.

In this section, we introduce preliminary results on singularities of the minimal model program, toric and cluster type log Calabi–Yau pairs, families of pairs, and dlt modifications.



 §.§ Singularities of the MMP



A pair (X,B) consists of a normal quasi-projective variety X and an effective -divisor B on X such that K_X+B is Q-Cartier. If {B_i | i ∈ I} are the prime components of B, then the strata of (X,B) are the irreducible components of the intersections ⋂_i ∈ J, J ⊂ I B_i.
    The index of a pair (X,B) is the smallest positive integer m such that m(K_X + B) is Cartier.




Let (X,B) be a pair. Given a projective birational morphism π Y → X from a normal quasi-projective variety and a prime divisor E on Y, the log discrepancy of (X, B) at E is 
    a_E(X, B)  1 - _E(π^*(K_X + B)).

    A pair (X,B) is terminal (resp. canonical, Kawamata log terminal (klt), lc canonical (lc)) if a_E(X, B) > 1 (resp. a_E(X, B) ≥ 1, a_E(X, B) >0 and ⌊ B ⌋ = 0, a_E(X, B) ≥ 0) for every exceptional divisor E over X.




Let (X,B) be a pair and f Y → X a birational morphism. The log pullback of B is the (not necessarily effective) Q-divisor B_Y defined by K_Y+B_Y = f^*(K_X+B) and f_* B_Y = B.




Let (X,B) be a log canonical pair. A divisor E over X is a log canonical place (resp. canonical place, non-canonical place, terminal place, non-terminal place) if a_E(X, B) = 0 (resp. a_E(X,B) = 1, a_E(X,B) < 1, a_E(X,B) > 1, a_E(X,B)≤ 1). The center of E on X is the closure of its image in X, and E is exceptional if its center on X is not a divisor.




Let ϕ Y  X be a rational map of normal quasi-projective varieties. We say ϕ contracts a Weil divisor E ⊂ Y if the center of E on X is not a divisor. We say ϕ extracts a Weil divisor F ⊂ X if ϕ^-1 contracts F.
The map ϕ is a birational contraction if it is birational and does not extract any divisors.




An lc pair (X,B) is divisorially log terminal (dlt) if the coefficients of B are at most one and if there is an open subset U ⊂ X such that  
  * U is smooth and B|_U is an snc divisor, and 
  * if E is a divisor over X with a_E(X, B) = 0, then the center of E on X intersects U. 

    If (X, B) is an lc pair, a Q-factorial dlt modification of (X,B) is a projective birational morphism π Y → X from a Q-factorial normal variety Y such that Ex(π) is a divisor, π only contracts log canonical places, and the log pullback (Y, B_Y) is dlt. A Q-factorial dlt modification exists for any lc pair by <cit.>.
    



Let X be a normal quasi-projective variety.
    
    
  * We say X is of Fano type if there exists an effective Q-divisor B such that (X, B) is a klt pair and -(K_X+B) is big and semiample.
    
  * A log Calabi–Yau (log CY) pair is an lc pair (X,B) such that K_X + B ∼_ Q 0.
    



X is Fano type if and only if there exists an effective Q-divisor B on X such that (X,B) is a klt pair and -(K_X+B) is ample <cit.>.
If X is projective, then by <cit.> an lc pair (X,B) is log CY if and only if K_X + B ≡ 0.


Two pairs (X_1, B_1) and (X_2, B_2) are crepant birational equivalent, written (X_1, B_1)≃_ cbir (X_2, B_2), if there exist proper birational morphisms f_i Y → X_i such that the log pullbacks of B_i on Y are equal.





 §.§ Toric log Calabi–Yau pairs

Now we define toric and cluster type log Calabi–Yau pairs.


A pair (X,B) is said to be toric
if X is a toric variety
and B is an effective torus-invariant divisor.



Recall that if X is a normal toric variety and B is the reduced sum  of the torus-invariant divisors, then K_X + B ∼ 0 <cit.> and the pair (X, B) is lc <cit.>.
In the case of log CY pairs, the following definitions generalize the notion of being toric.


Let (X,B) be a log CY pair.

    
  * We say that (X,B) is toric
if X is a normal toric variety and B is the reduced
sum of the torus-invariant divisors.[For log CY pairs, this condition is equivalent to (X,B) being a toric pair as in Definition <ref>.]

  * We say that (X,B) is of cluster type if there exists a crepant birational 
map ϕ (^n,Σ^n) (X,B) such that 𝔾_m^n ∩ Ex(ϕ)
contains no divisors on X.

  * We say that (X,B) is log rational if (X,B)≃_ cbir (T,B_T) for T a normal projective toric variety
and B_T the reduced sum of the torus-invariant divisors;
this condition is equivalent to (X,B) ≃_ cbir (^n,Σ^n).

In a similar vein, we say that an algebraic variety X is of cluster type (resp. log rational) if it admits a boundary divisor B for which (X,B) is a cluster type 
(resp. log rational) log CY pair.


Every toric log CY pair (X,B) is of cluster type, as it admits a torus-equivariant 
crepant birational map (^n,Σ^n) (X,B) that is an isomorphism
on 𝔾_m^n.
Thus, the following implications hold for a log CY pair (X, B):

    (X,B) is toric(X,B) is of cluster type(X,B) is log rational.


The reverse implications do not hold; see Example <ref> and <cit.>.
If (X,B) is a log CY pair of index one, then (X,B) admits a toric
model if and only if the pair (X,B) has birational
complexity zero <cit.>.
Since crepant birational equivalences preserve the indices of log CY pairs (see, e.g., <cit.>),
any log rational log CY pair necessarily has index one:

If (X,B) is a log CY pair that is log rational,
then K_X+B∼ 0. In particular, B is a reduced Weil divisor.


We now state several results about toric log CY pairs that we will use in the proofs of Theorems <ref> and <ref>. First, we recall some invariants of pairs that
Brown–McKernan–Svaldi–Zong <cit.> used to characterize toric log CY pairs:


Let (X,B) be a pair.


  * A decomposition Σ of B is a finite formal sum ∑_i=1^k α_i B_i = B where each α_i ≥ 0 and B_i is a reduced (not necessarily irreducible) effective divisor. 
  * The Picard rank of Σ is ρ(Σ)(span{B_i | 1 ≤ i ≤ k}) where the span is inside _ Q X  X⊗_ Z Q. The norm of Σ is |Σ| ∑_i=0^k α_i. The complexity of the decomposition (X,B;Σ) is c(X,B;Σ)  X + ρ(Σ) - |Σ|.



If (X,B) is a toric log CY pair and Σ is the decomposition of B given by the sum of its prime divisors, then c(X,B;Σ) = 0 (see, e.g., <cit.>).

 Let (X,B) be a log CY pair and Σ a decomposition of B. Then c(X,B;Σ) ≥ 0.
If c(X,B;Σ) < 1, then there exists a divisor D ≥⌊ B ⌋ such that (X, D) is a toric log CY pair, and all but ⌊ 2 c(X,B;Σ) ⌋ components of D are elements of the set {B_i | 1 ≤ i≤ k}.
In particular, (X, ⌊ B⌋) is a (not necessarily log CY) toric pair.


The following lemma is well known to the experts (see, e.g., <cit.>).


Let (X, B) be a projective toric log CY pair, and
let π X Y be a birational contraction
to a projective variety.
Then (Y, π_* B) is a toric log CY pair,
and π is a toric birational map.


The following result, which shows that certain birational modifications of toric log CY pairs are toric, will be useful in the proofs of both Theorems <ref> and <ref>.

Let (X,B) be a projective toric log CY pair,
let p Y X be a birational contraction that only contracts non-canonical places of (X,B),
and let (Y,B_Y) be the log pullback of (X,B) to Y.
Then (Y,B_Y) is a toric log CY pair, and Y X is a toric projective birational morphism.



The pair (X,B) has index one by <cit.>,
so the assumption on p implies it only contracts log canonical places of (X,B).
In particular, we may find a -factorial dlt modification (Z,B_Z)→ (X,B)
such that Z Y is a birational contraction.
Let Σ (resp. Σ_Z) be the decomposition of B (resp. B_Z) given by the sum of its prime divisors.
Then c(X,B;Σ)=0 since (X,B) is toric. Since
ρ(Σ_Z)=ρ(Σ)+ρ(Z/X)
and |Σ_Z|=|Σ|+ρ(Z/X),
we get 
c(Z,B_Z;Σ_Z)=0,
so (Z,B_Z) = (Z, ⌊ B_Z ⌋) is a toric log CY pair by Theorem <ref>.
Applying Lemma <ref> to Z  Y  X, we conclude that (Y,B_Y) is a toric log CY pair, and hence that Y X is a toric birational contraction.




 §.§ Families of varieties and pairs


Next, we will consider relative versions of Definition <ref>. A projective contraction is a morphism f X → T of normal quasi-projective varieties such that f_*  O_X =  O_T. A fibration is a projective contraction whose general fiber is positive dimensional.


    Let (X,B = ∑α_i B_i) be a pair and f X → T a projective contraction. For a closed point t∈ T, the log fiber is (X_t, B_t ∑_B_i ⊅X_tα_i B_i|_X_t).
    If X_t is normal and is not contained in any prime divisor in the support of B, then (X_t, B_t) is a pair.

A family of varieties is a flat projective contraction f X → T of normal quasi-projective varieties.
A family of pairs is a family of varieties X → T and a pair (X, B) such that the log fiber over any closed point is a pair.



    
  * A family of Fano varieties is a family of varieties such that the fiber over every closed point is a Fano variety.
    
  * A Fano fibration is a projective contraction X → T such that -K_X is ample over T.
    
  * A Fano type morphism is a projective contraction X → T such that there exists a
    Q-divisor B on X with (X,B) a klt pair, B big over Z, and K_X + B ∼_T,  0.
    
  * A family of log CY pairs is a family of pairs such that the log fiber over any closed point is a log CY pair.
    
  * A log CY fibration is a projective contraction X→ T such that there exists a -divisor B on X with (X,B) an lc pair and K_X+B∼_T, 0.




Note that a family of Fano varieties is not necessarily a Fano fibration, but it becomes a Fano fibration after a finite étale base change, since the geometric generic fiber is Fano. On the other hand, a Fano fibration is not necessarily a family of Fano varieties over T, but it is over some dense open subset of the base. The same properties hold for log CY pairs.


Let X→ T be a Fano type morphism.
Let Y be a normal quasi-projective variety and Y→ X a projective birational morphism
that only contracts non-terminal places of X.
Then, the composition Y→ T is a Fano type morphism.



Let Δ be a boundary on X such that
(X,Δ) is klt and -(K_X+Δ) is big and semiample over T.
The log pullback Δ_Y of Δ is effective
as Y→ X only
contracts non-terminal places of (X,Δ),
so (Y,Δ_Y) is a klt pair.
Since -(K_Y+Δ_Y) is big and semiample over T,
the morphism Y→ T is of Fano type.


The following lemma is well known (see, e.g., <cit.>).


Let X→ T be a Fano type morphism, and
let (X,B) be a pair that is log CY over T, i.e., K_X + B ∼_T,  Q 0.
Let Y be a normal quasi-projective variety and Y→ X a projective birational morphism
that only contracts non-canonical places of (X,B).
Then, the composition Y→ T is a morphism of Fano type.


For a rational map π X  Z of normal varieties, we define the pullback of divisors as follows. Let X be a normal variety that resolves π, with projections p_1X→ X and p_2X→ Z. Let D_Z be a Q-Cartier divisor on Z, and assume m D_Z is Cartier. Then we define the Q-divisor 
    π^* D_Z 1/mp_1_* p_2^* (mD_Z),

where p_2^* (mD_Z) is the Weil divisor associated to the Cartier divisor corresponding to m D_Z under the isomorphism 

    H^0(Z,  O_Z(mD))  H^0(Z, p_2_* p_2^* O_Z(mD_Z)) = H^0(X, p_2^* O_Z(mD_Z)).


If D_X is a prime divisor on X, we define its pushforward π_* D_X to be its strict transform on Z. We extend π_* linearly to Weil divisors on X. If π is a birational contraction, then the equality π_* π^* D = D holds as Q-divisors on Z.
Furthermore, if π is a birational contraction, then π_* induces a homomorphism X → Z.

We now introduce some notions of class groups in families.


Let f X→ T be a fibration of Q-factorial normal varieties.
We say that f has surjective class group restrictions (resp. injective class group restrictions, isomorphic class group restrictions)
if the restriction homomorphism

    Cl_ Q(X/T)  ( X / f^*  T) ⊗_ Z Q → Cl_ Q(X_t)  (X_t) ⊗_ Z Q

is surjective (resp. injective, an isomorphism) for every closed point t∈ T.



The property of injective class group restrictions descends under birational contractions:


Let ϕ X → T and ψ Z → T be fibrations of Q-factorial varieties
such that the fibers X_t and Z_t are normal for every t ∈ T.
Assume there is a birational contraction
π X Z over T.
If ϕ has injective class group restrictions,
then ψ also
has injective class group restrictions.



Let D be a Q-divisor on Z
for which D|_Z_t∼_ 0 for some t∈ T.
The pullback π^*D to X satisfies
π^*D|_X_t∼_ 0,
so π^*D∼_,T 0
because X→ T has injective class group restrictions.
Since π is a birational contraction, we conclude
D = π_*π^*D ∼_,T 0.


The following properties of class groups of fibers of Fano type morphisms will be useful.


    Let f X → T be a Fano type morphism of Q-factorial varieties.
    
        
  *  Let t_0∈ T be a closed point in the smooth locus of T. There is a smooth open neighborhood t_0∈ U⊂ T such that for any divisor D on X over T, if D|_X_t_0≡ 0, then D is Q-linearly trivial over an open neighborhood of t_0.[That is, there exists an open neighborhood U ∋ t_0 such that D|_X_U∼_ Q 0 in (X_U/U)  X_U / f|_X_U^* U.]
        
  *  There exists a nonempty open subset V ⊂ T such that
        f|_X_V X_V → V has injective class group restrictions.
        
  *  If f has surjective class group restrictions, then f has isomorphic class group restrictions
        over a nonempty open subset of T.
    



    First, we note that for any divisor D on X over T that is numerically trivial on X_t_0, there is an open neighborhood U_D∋ t_0 over which D|_X_U_D≡ 0. Indeed, over any affine open subset of T, we may run a D-MMP for X that terminates with a good minimal model by <cit.>, and the assumption that D is numerically trivial on X_t_0 implies that this MMP is trivial on an open neighborhood of t_0. Since f is of Fano type and D is numerically trivial over this open neighborhood, it is Q-linearly trivial over this open neighborhood.
    
    For (<ref>), since X→ T is Fano type, _ Q(X/T) ≅_ Q(X/T)/≡_ Q, T is finitely generated (see <cit.>).
    Thus, after replacing T by an open neighborhood of t_0 finitely many times, the conclusion holds.

    For (<ref>), let t_0∈ T, and let t_0 ∈ U_0 be the open neighborhood from (<ref>). Then (_ Q(X/T) →_ Q(X_t)) ⊆(_ Q(X/T) →_ Q(X_t_0)) for any t∈ U_0. If the containment is strict for some t_1∈ U_0, apply (<ref>) to find an open neighborhood t_1 ∈ U_1 ⊂ U_0. Repeating this process, we obtain a sequence (_ Q(X/T) →_ Q(X_t_i+1)) ⊆(_ Q(X/T) →_ Q(X_t_i)) of subspaces of the finite-dimensional Q-vector space _ Q(X/T). This must stabilize after a finite number of steps, so after shrinking T a finite number of times, we obtain the open subset V in (<ref>).
    Finally, (<ref>) is immediate from (<ref>).




 §.§ Properties of cluster type log Calabi–Yau pairs and dlt modifications


Now we focus on log CY pairs of cluster type and prove several results that we will use in the proof of Theorem <ref>.
The next lemma is one of the reasons for the terminality assumption the theorem.


Let (X,B) be a pair of index one,
and let (X,Δ) be a terminal pair.
For every ϵ∈ (0,1),
the exceptional non-terminal places of (X,(1-ϵ)B+ϵΔ)
are log canonical places of (X,B).



If E is an exceptional non-terminal place
of (X,(1-ϵ)B+ϵΔ), then

    a_E(X,(1-ϵ)B+ϵΔ) = 
    (1-ϵ)a_E(X,B) + ϵ a_E(X,Δ) ≤ 1.

As a_E(X,Δ)>1, we have ϵ a_E(X,Δ)>ϵ
and so (1-ϵ)a_E(X,B)< 1-ϵ.
Thus, a_E(X,B)<1.
Since (X,B) has index one, we get a_E(X,B)=0,
so E is a log canonical place of (X,B).


Next, we show that a dlt modification of a cluster type log CY pair is also of cluster type.


Let (X,B) be a log CY pair of cluster type.
Let ϕ Y  X be a birational map
that only extracts log canonical places of (X,B),
and only contracts non-terminal places of (X,B).
Let (Y,B_Y) be the log pullback of (X,B) to Y.
Then (Y,B_Y)
is a log CY pair of cluster type.



First, the assumption that ϕ only contracts non-terminal places implies that B_Y is effective, so (Y, B_Y) is a pair.
Let π (^n,Σ^n) (X,B) be a crepant birational map
such that 𝔾_m^n ∩ Ex(π) has codimension at least two on the torus.
By the assumption that ϕ only extracts log canonical places, the divisorial part of the  exceptional locus of ϕ^-1 X Y
is contained in supp(B).
We conclude that 
ϕ^-1∘π (^n,Σ^n) (Y,B_Y) is a crepant birational map for which 
𝔾_m^n ∩ Ex(ϕ^-1∘π) has codimension at least two in the torus.
So (Y,B_Y) is of cluster type.


The property of being a cluster type log CY pair also descends under dlt modifications.


Let (Y,B_Y) be a log CY pair
of cluster type.
If ϕ Y X is a birational contraction that only contracts 
log canonical places of (Y,B_Y),
then (X,ϕ_* B_Y) is a log CY pair of cluster type.



Write B ϕ_* B_Y.
First note that ϕ (Y,B_Y)  (X, B) is crepant birational by the negativity lemma.
Let ψ (^n,Σ^n) (Y,B_Y) be a crepant birational map
such that 𝔾_m^n∩ Ex(ψ)
has codimension at least two in 𝔾_m^n.
Then, as ϕ only contracts log canonical places of (Y,B_Y), the composition 
ϕ∘ψ (^n,Σ^n) (X,B) is a crepant birational map
such that 𝔾_m^n ∩ Ex(ϕ∘ψ) has codimension at least two in 𝔾_m^n.


Next, we construct dlt modifications with certain desirable properties. The nice dlt modifications constructed in the following lemmas will be useful in giving another characterization of cluster type log CY pairs (Lemma <ref>) and in proving Theorem <ref>. First, we need some definitions.


Let (X,B) be a dlt pair 
and X^ snc be the largest open subset of X
on which the pair is simple normal crossing.
Let B^ snc be the restriction of B to X^ snc.


  * A formally toric blow-up
of (X^ snc,B^ snc)
is a blow-up Y^ snc→ X^ snc that is a formally toric morphism over any closed point of (X^ snc,B^ snc) (see <cit.> for more details).
We say that a formally toric blow-up is a formally toric dlt blow-up
if the log pullback of (X^ snc,B^ snc) to Y is a dlt pair.

  * A formally toric dlt blow-up of (X,B) is a dlt modification
(Y,B_Y)→ (X,B) that restricts to a formally toric blow-up 
of (X^ snc,B^ snc).
We say that a formally toric blow-up
is a blow-up of a stratum if it is induced by the blow-up of the 
reduced scheme structure of a stratum of (X^ snc,B^ snc).

    



<cit.> shows
that if (X,B) is a dlt pair, then every formally toric dlt blow-up of
(X^ snc,B^ snc) extends to a formally toric blow-up
of (X,B).
Thus, every stratum of (X,B) induces a formally toric blow-up of the stratum, and this formally toric dlt blow-up of (X,B) is unique up to a small birational transformation over the base.
    



Let (X,B) be a log canonical pair,
and let E⊂ X be an effective Q-Cartier Q-divisor that 
has no common components with B.
Then there is a Q-factorial dlt modification ϕ (Y,B_Y)→ (X,B) such that the strict transform ϕ^-1_*E of E on Y
contains no log canonical centers of (Y,B_Y). Furthermore, the pair (Y,B_Y+ ϵϕ_*^-1 E) is dlt for sufficiently small ϵ > 0.

If moreover (X,B) is a dlt pair, then ϕ may be obtained by a sequence of formally toric blow-ups of strata of (X,B). 



Consider the pair (X,B+ϵ E), which is not necessarily log canonical.
By linearity of log discrepancies with respect to the boundary divisor,
for ϵ>0 small enough, every non-lc center of (X,B+ϵ E)
is a log canonical center of (X,B).
By <cit.>
there exists a Q-factorial dlt modification
ϕ (Y,B_Y+ϵϕ^-1_* E)→ (X,B+ϵ E) where
B_Y is the sum of
ϕ^-1_* B and the reduced exceptional divisor of ϕ (see <cit.>).
Then (Y, B_Y) is dlt, and
by the previous considerations, we have 
ϕ^*(K_X+B)=K_Y+B_Y.
Since (Y,B_Y+ϵϕ^-1_* E)
is log canonical, we know that  ϕ^-1_* E contains no log canonical centers of (Y,B_Y).

Now, assume in addition that (X,B) is dlt.
By the previous paragraph,
there exists a dlt modification ϕ_0 (Y_0,B_Y_0)→ (X,B)
such that ϕ_0^-1_*E
contains no log canonical centers of (Y_0,B_Y_0).
We first argue that the dlt modification (Y_0,B_Y_0) induces a formally toric dlt blow-up
(Y_0^ snc,B_Y_0^ snc)→ (X^ snc,B^ snc).
Locally at every closed point, the pair (X^ snc,B^ snc) has local complexity zero <cit.>.
The morphism (Y_0^ snc,B_Y^ snc_0)→ (X^ snc,B^ snc) only contracts
divisors with coefficient one, so the complexity of this morphism is zero at every closed point of X^ snc (see <cit.>).
Hence, <cit.> implies that the dlt modification
(Y_0^ snc,B_Y^ snc_0)→ (X^ snc,B^ snc) is formally toric over the base.
That is, ϕ_0 is a formally toric dlt blow-up
along some ideal sheaf whose support is contained in supp(B^ snc).
Now we will construct Y as be a higher model of Y_0^ snc that is obtained from (X,B) by a sequence of formally toric dlt blow-ups along strata.

We first construct this model over formal neighborhoods in X^ snc.
Here the morphism Y_0^ snc→ X^ snc
corresponds to unimodular fan refinements of unimodular cones
of (X^ snc,B^ snc) <cit.>.
Every refinement of a unimodular cone σ is refined by a sequence of stellar subdivisions of σ <cit.>.
Hence, over formal neighborhoods in X^ snc, there is a formally toric dlt blow-up 
(Y^ snc,B_Y^ snc)→ (X^ snc,B^ snc)
that is obtained by successively blowing up strata of (X^ snc, B^ snc)
<cit.>
and such that there is a projective birational
morphism Y^ snc→ Y_0^ snc.
Furthermore, additional blow-ups along strata of (X^ snc,B^ snc) result in a higher model that is still dlt and formally toric over (X^ snc,B^ snc).
Now, consider the locally closed decomposition X^ snc = _i ∈ I Z_i defined by the strata of B. For each i∈ I, the above procedure produces the same sequence of blow-ups over any closed point of the locally closed subset Z_i. Since the index set I is finite,
we may apply the above procedure to each Z_i (performing additional blow-ups along strata if necessary) to get a sequence of blow-ups of strata of (X^ snc,B^ snc), which yields a formally toric dlt blow-up (Y^ snc,B_Y^ snc)→ (X^ snc,B^ snc) that factors through a projective birational
morphism Y^ snc→ Y_0^ snc.
By Remark <ref>, this extends to a formally toric dlt blow-up
(Y,B_Y)→ (X,B).
By construction, the induced birational map
ψ Y Y_0 is a birational contraction
and Ex(ψ) contains no log canonical centers of (Y,B_Y).
Then, the strict transform of E on Y contains no log canonical centers of (Y,B_Y).


We can now give the following equivalent characterization of cluster type log CY pairs.


A log CY pair (X,B)
is of cluster type if and only if 
there exists a Q-factorial dlt modification (Y,B_Y)→ (X,B)
with a crepant birational contraction
π (Y,B_Y)  (^n,Σ^n).
Furthermore, if E  Ex(π)∖ supp(B_Y), we may assume that
(Y,B_Y+ϵ E) is a dlt pair for ϵ > 0 small enough.



First, assume that (X,B) is of cluster type.
Then there exists a crepant birational map
(^n,Σ^n) (X,B)
that only contracts log canonical places.
In particular, there exists a Q-factorial dlt modification (Z,B_Z)→ (X,B)
such that the composition p Z^n is a birational contraction. Let E_Z  Ex(p)∖ supp(B_Z).
Applying Lemma <ref> to (Z, B_Z) and E_Z
yields a dlt modification q (Y,B_Y)→ (Z,B_Z) such that (Y,B_Y+ ϵ q^-1_*E_Z) is dlt for ϵ>0 small enough.
Denote the composition by π p ∘ q Y  P^n.
Then q^-1_*E_Z= Ex(π)∖ supp(B_Y),
so (Y, B_Y) → (X,B) is the desired dlt modification.

Now assume (X,B) admits a Q-factorial dlt modification ϕ (Y,B_Y)→ (X,B)
with a crepant birational contraction π (Y,B_Y) (^n,Σ^n).
Then (X,B) is of cluster type by Lemma <ref>.
Furthermore, if E =  Ex(π)∖ supp(B_Y), then by Lemma <ref> we may replace (Y,B_Y) with a higher -factorial dlt modification
such that (Y,B_Y+ϵ E) is dlt for ϵ>0 small enough.


The next lemma will be used to ensure that certain linear combinations of pairs are terminal on higher dlt modifications.


Let (X,B) be a dlt pair of index one.
Let (X,Δ) be a terminal pair, and assume that the prime components of Δ are Q-Cartier.
Let (Z,B_Z)→ (X,B) be any dlt modification.
Then for any sufficiently small ϵ > 0, there exists a Q-factorial dlt modification (Y,B_Y)→ (Z,B_Z)
such that the log pullback of
(X,(1-ϵ)B+ϵΔ) to Y
is a terminal pair.



Let D be the (reduced) sum of the prime components of Δ
that are not components of B.
By Lemma <ref>, there is a Q-factorial dlt modification ϕ (W,B_W)→ (Z,B_Z)
such that
ϕ^-1_* D
contains no log canonical centers of (W,B_W).
For sufficiently small ϵ > 0,
every divisor contracted by W→ X
is a non-terminal place of (X,(1-ϵ)B+ϵΔ),
so the log pullback (W,Γ_W) of (X,(1-ϵ)B+ϵΔ) to W is a pair; furthermore, the pair (W,Γ_W) is klt.

By Lemma <ref>,
every exceptional non-terminal place of (X,(1-ϵ)B+ϵΔ)) is a log canonical place of (X,B).
In particular, every exceptional non-terminal place of (W,Γ_W) is a log canonical place of (W,B_W),
and the log canonical centers of (W,B_W) are precisely its proper strata.
By construction, the components of Γ_W that are not contained in B_W are precisely the components of ϕ^-1_* D.
So, on a neighborhood of
the generic point of every proper stratum of (W,B_W),
the support of Γ_W is contained
in the support of B_W.
Since (W,B_W) is log smooth along the generic point of each stratum,
we conclude that (W,Γ_W) is log smooth around every non-terminal center.
Therefore, we may obtain a terminal model of (W,Γ_W)
by successively blowing up strata of (W,B_W) (see <cit.>).
Hence, the composition (Y,B_Y)→ (W,B_W)→ (Z,B_Z) is a Q-factorial dlt modification 
for which the log pullback of
(X,(1-ϵ)B+ϵΔ) is terminal.


Finally, we will apply the following two technical lemmas to certain fibrations in Section <ref> to show that, after shrinking the base, the outcomes of certain MMPs will have nice fibers.


Let (Y,Γ) be a terminal pair and f (Y,Γ)→ T a log Calabi-Yau fibration. 
Let U  U_1 ∩ U_2 ∩ U_3 be the intersection of the following dense open subsets of T:

    
  * the smooth locus U_1 ⊂ T, 
    
  * an open set U_2 over which all log fibers (Y_t,Γ_t) are terminal and Γ_t = Γ|_Y_t, and
    
  * the complement U_3 of the boundary divisor and the moduli divisor induced by the canonical bundle formula
    on T by (Y,Γ). 

Let E be an effective -Cartier Q-divisor on Y_U
that contains no fiber of f.
Then for δ > 0 small enough and
for any sequence Y_U W of steps of the (K_Y_U+Γ_U+δ E)-MMP over U,
the following conditions hold:

    
  * the fibers of Y_U→ U are normal, 
    
  * the fibers of W→ U are normal, 
    
  * for every t∈ U, the induced birational map Y_t W_t is a contraction, and 
    
  * if Γ_W is the pushforward of Γ to W, then the log fibers of (W,Γ_W)→ U are terminal. 




Let d be the dimension of U, and let
t∈ U be a closed point.
Then (<ref>) holds by <ref>; furthermore, if H_1,…,H_d are general hyperplanes on U containing t, the pair
(U,H_1+…+H_d;t) is log smooth and has t as a log canonical center.
Consider the pair

    (Y,Γ+f^*H_1+…+f^*H_d).

This pair is log canonical over a neighborhood of t by <ref> and the canonical bundle formula (see, e.g., <cit.>).
The pair (Y_t,Γ_t) is terminal by <ref>,
so Y_t is a minimal log canonical center of (<ref>). Further (Y_t,Γ_t) is the pair induced by adjunction to the minimal log canonical center:
indeed, if (Y_t,Γ_t) is the pair obtained by adjunction of (<ref>) to Y_t, then (Y_t,Γ_t) is log CY and Γ_t≥Γ_t, so Γ_t=Γ_t.

In what follows, we will show that there is an open neighborhood of t such that, for sufficiently small δ > 0 depending on t, the conclusion holds over this neighborhood of t. Since U is compact, we can find δ that works for all t∈ U.

Let E_t denote the restriction of E to Y_t.
By assumption, E contains no log canonical centers of (<ref>) over a neighborhood of t.
Therefore, for δ>0 small enough, the pair

    (Y,Γ+f^*H_1+…+f^*H_d+δ E)

is also log canonical over a neighborhood of t, 
and the pair (Y_t,Γ_t+δ E_t), which is obtained by adjunction
to the minimal log canonical center,
is terminal.
Let ϕ Y_U  W be a sequence
of steps of the (K_Y_U+Γ_U+δ E)-MMP over U.
Then, we have a commutative diagram

    (Y_U,Γ_U+f^*H_1+…+f^*H_d+δ E)@–>[r]^-ϕ[d]_-f    
    (W,Γ_W+f'^*H_1+…+f'^*H_d+δ E_W) [ld]^-f'
     
    T

where Γ_W (resp. E_W) is the pushforward
of Γ_U (resp. E) to W.
The pair 

    (W,Γ_W+f'^*H_1+…+f'^*H_d+δ E_W)

is log canonical over a neighborhood of t,
and W_t is a minimal log canonical center.
By <cit.>, this implies that (<ref>) holds.

It remains to show (<ref>) and (<ref>). First, by adjunction, the induced birational map

    ϕ_t (Y_t, Γ_t+δ E_t)  (W_t,Γ_W_t+δ E_W_t)

between minimal log canonical centers of (<ref>) and (<ref>)
is (K_Y_t+Γ_t+δ E_t)-negative,
so the negativity lemma applied to a resolution of ϕ_t shows that

    a_F_t(W_t,Γ_W_t+δ E_W_t) ≥
    a_F_t(Y_t,Γ_t+δ E_t)

for any prime divisor F_t.
We now show (<ref>).
By contradiction, suppose ϕ_t is not a birational contraction, so it extracts some prime divisor F_t. As F_t is exceptional over Y_t, the inequality (<ref>) implies that
F_t appears in Γ_W_t+δ E_W_t
with negative coefficient,
contradicting the fact that (W_t,Γ_W_t+δ E_W_t)
is a pair.
This proves (<ref>).
Finally, the inequality (<ref>) and the terminality of (Y_t,Γ_t+δ E_t) imply that (W_t,Γ_W_t+δ E_t) is terminal and hence that (W_t,Γ_W_t) is terminal, so (<ref>) holds.



Let (Y,B) be a dlt pair and f (Y,B)→ T a log Calabi–Yau fibration.
Let U  U_1 ∩ U_2 ∩ U_3 be the intersection of the following dense open subsets of T:

    
  * the smooth locus U_1 ⊂ T, 
    
  * an open set U_2 over which the log fibers of (Y,B)→ T are dlt and B_t = B|_Y_t, and 
    
  * the complement U_3 of the support of the boundary divisor and the moduli divisor induced by the canonical bundle formula on T by (Y,B). 

Let E be an effective ℚ-Cartier Q-divisor E on Y_U that does not contain any fiber of f
and does not contain any irreducible component of the restriction of 
any log canonical center of (Y,B) to a fiber of f.
Then for δ > 0 small enough and
for any sequence
Y_U  W of steps of the (K_Y_U+B_U+δ E)-MMP over U,
the normalizations of the log fibers of (W,B_W)→ U are lc.



Let d be the dimension of U, and let t∈ U be a closed point.
As in the proof of Proposition <ref>, we may work on an open neighborhood of t.
Using <ref>,
let (U,H_1+…+H_d;t) be a log smooth pair
with t a log canonical center,
as in the proof of Lemma <ref>.
By the canonical bundle formula
and <ref>, the pair 

    (Y,B+f^*H_1+…+f^*H_d)

is log canonical over a neighborhood of t,
and Y_t is a log canonical center.

Since (Y,B) and (Y_t, B_t) are dlt, the lc centers of (Y_t, B_t) are irreducible components of
restrictions of lc centers of (Y,B).
So, by assumption, E contains no lc centers of (Y_t,B_t).
Then inversion of adjunction <cit.> and the assumption that Y_t⊄E imply that E contains no
log canonical centers of (Y,B+f^*H_1+…+f^*H_d) near t.
Thus, for sufficiently small δ > 0,
the pair
(Y,B+f^*H_1+…+f^*H_d+δ E)
is log canonical over a neighborhood of t.

Let Y_U W be a sequence
of steps of the (K_Y_U+B_U+δ E)-MMP
over U,
and let f' W→ U be the induced fibration.
Let B_W (resp. E_W) 
be the pushforward of B (resp. E) to W.
Then, over a neighborhood of t, the pair 
(W,B_W+f'^*H_1+…+f'^*H_d+δ E_W)
is log canonical,
and hence so is

    (W,B_W+f'^*H_1+…+f'^*H_d).
 
Since the log fiber (W_t,B_t) is
the pair obtained by adjunction
of (<ref>) to W_t,
we conclude by <cit.> that
its normalization is lc.




§ TORIC LOG CALABI–YAU PAIRS IN FAMILIES


In this section, we prove that in a family of log Calabi–Yau pairs,
the property of being toric is constructible. Later, in Section <ref>, we will use this result to prove Theorem <ref>.


Let f (𝒳,ℬ) → T be a family of log
Calabi–Yau pairs of index one, and assume that X_t is of Fano type for every closed point t∈ T. Then, the set 

    T_ toric{closed t ∈ T | (𝒳_t,ℬ_t)  is a toric log CY pair}

is a constructible subset of T. 


First, we prove that for certain nice fibrations,
the existence of a toric log fiber implies that every log fiber is toric. We will use the characterization of toric pairs proven by <cit.>, which uses complexity (see Definition <ref>).

 Let X→ T be a projective contraction of Q-factorial normal quasi-projective varieties, 
(X,B) a pair, and t_0∈ T a closed point.
Assume the following conditions are satisfied:


  * X_t_0 =  X_t for t in T, 

  * B is reduced and contains no fibers of X → T, 

  * the log fibers of (X,B)→ T are log canonical pairs, 

  * the restriction of B to B|_X_t_0 induces a bijection between prime components, 

  * X → T has injective class group restrictions (Definition <ref>), and 

  * (X_t_0, B_t_0) is a toric log CY pair. 

Then all log fibers of (X,B) → T are toric log CY pairs.



By assumption (<ref>), the restriction of K_X+B to X_t_0 is -linearly trivial.
By (<ref>), we conclude that K_X+B is -linearly trivial on every closed fiber X_t.
In particular, by (<ref>) and (<ref>), the log fiber (X_t,B_t) is a log CY pair for every t∈ T.


Now write b for the number of prime components B_t_0,1,…,B_t_0,b of B_t_0, and consider the decomposition Σ_t_0∑_i=1^b B_t_0, i of B_t_0. Since (X_t_0, B_t_0) is a toric pair, we have the equality b =  X_t_0 + ρ(Σ_t_0) (see, e.g., <cit.>).
Now, for each 1≤ i ≤ b, let Γ_i be the prime component of B restricting to B_t_0, i (using (<ref>)). Let t∈ T be a closed point,
and consider the decomposition of B_t given by Σ_t ∑_i=1^b Γ_i|_X_t = B_t.
Then |Σ_t| = b, X_t =  X_t_0 by (<ref>), and ρ(Σ_t)  = ρ(Σ_t_0) by  (<ref>), so  


    c(X_t, B_t; Σ_t) = c(X_t_0, B_t_0; Σ_t_0) = 0.
 

Hence (X_t, B_t) = (X_t, ⌊ B_t ⌋) is a toric log CY pair by Theorem <ref>.


 Let (X,B)→ T be a log Calabi–Yau fibration of index one over T, i.e., K_X+B∼_T 0. Assume X→ T is Fano type.
Then there is an open subset U of T such that if the log fiber (X_t,B_t) is a toric CY pair for some closed t∈ U, then every log fiber over 
U is a toric log CY pair.



    Let T'→ T be a finite Galois cover such that for the pullback (X_T', B_T'), the prime components of B_T' restrict to prime divisors on general fibers of X_T'→ T'. Let (X”, B”)→ (X_T', B_T') be a Q-factorial dlt modification, which exists by <cit.>. We get a commutative diagram
    
    (X,B) [d]     (X_T', B_T') [d] [l]     (X”,B”)  [l] [ld]^-ψ
     T     T' [l]_-π.

    Let U'⊂ T' be a nonempty open subset over which (X”,B”) → U' satisfies conditions (<ref>), (<ref>), and (<ref>) of Proposition <ref>. By shrinking U' if necessary, we may further assume that (X”,B”) → (X_T', B_T') induces dlt modifications on log fibers over U'.
    Lemma <ref> shows that X”|_U'→ U' is Fano type. We may assume by Lemma <ref>(<ref>) that Proposition <ref>(<ref>) holds over U'.
    Let U⊂ T be an open subset whose preimage in T' is contained in U'. We may assume π is étale over U.
    Then ψ (X”, B”) → U' satisfies conditions (<ref>)–(<ref>) of Proposition <ref>.
    
    Now assume (X_t, B_t) is toric log CY for some closed t∈ U, and let t' ∈ T' be a preimage of t. Then (X'_t', B'_t') is toric log CY. A dlt modification of a toric log CY pair is also toric log CY by Lemma <ref>. Hence, the log CY pair (X”_t', B”_t') is toric.
    
    All log fibers of ψ over U' are toric log CY pairs by Proposition <ref>, so all fibers of (X_U', B_U') → U' are toric by Lemma <ref>. Hence, all log fibers of (X,B) → T over U are toric.


   
    We proceed by induction on T. If T = 0 there is nothing to show. Next, assume T ≥ 1.
    As the geometric generic fiber is an index one log Calabi–Yau pair on a Fano type variety, we may take a dominant finite étale morphism T' → T such that the base change (𝒳_T', ℬ_T') → T' is a log CY fibration and 𝒳_T'→ T' is a Fano type morphism.
    Let U' ⊂ T' be the open subset of Theorem <ref>, and let U ⊂ T be its image.
    Then T'_toric∩ U' is a constructible subset of T' by Theorem <ref>, so its image T_toric∩ U in T is also constructible. Since T_toric∩ (T∖ U) is constructible by induction, this completes the proof.




§ CLUSTER TYPE LOG CALABI–YAU PAIRS IN FAMILIES


In this section, we prove that in a family of terminal log Calabi–Yau pairs,
the property
of being cluster type is constructible.
In Section <ref>, we will use this to prove Theorem <ref>.


Let f (𝒳,ℬ)→ T 
be a family of log Calabi–Yau pairs of index one,
and assume that X_t is a -factorial terminal Fano variety for every closed point t∈ T.
Then, the set 

    T_ cluster{closed t∈ T | (𝒳_t,ℬ_t)
     is a log CY pair of cluster type}

is a constructible subset of T.


To prove Theorem <ref>, we will use the characterization of cluster type pairs in Lemma <ref>,
which shows that (X,B) is cluster type if and only if there is a Q-factorial dlt modification (Y, B_Y) that admits a crepant birational contraction ϕ (Y, B_Y)  ( P^n, Σ^n).

Theorem <ref> will follow from Theorem <ref>. The idea of the proof of Theorem <ref> is as follows.
Assume the log fiber over t_0 is cluster type.
Then, after taking an appropriate étale cover of T
and suitable higher dlt modifications of (Y_t_0, B_Y,t_0), we may spread out ϕ_t_0 to a family of crepant birational contractions ϕ_t  (Y_t, B_t)  (W_t, B_W, t) over an open neighborhood of a preimage of t_0. Using Proposition <ref>, we show that the log CY pairs (W_t, B_W, t) are toric. Hence, the log fibers of (𝒳,ℬ) are cluster type over an open neighborhood of t_0.

In order to spread out the birational contraction ϕ_t_0, we will need to extend certain divisors over a family.
An important ingredient for this is the following lemma
on invariance of sections
for suitable dlt modifications of Fano type morphisms.
This lemma extends <cit.>.


Let X→ T be a Fano type morphism of Q-factorial varieties.
Let (X,B) be a

dlt pair that is log CY of index one over T.
Assume that for every closed point t∈ T, each proper stratum of (X,B) restricts to a proper stratum of (X_t,B_t).
Assume there is a boundary Δ for which (X,Δ) is terminal
and log CY over T.
Then, there exists a non-empty affine open subset U⊆ T
satisfying the following condition:
for any dlt modification π (Y,B_Y)→ (X,B),
any Cartier divisor D on Y, and every closed point t∈ U,
we have a surjective homomorphism

    H^0(Y,𝒪_Y(mD)) ↠ H^0(Y_t,𝒪_Y_t(mD|_Y_t)),

for every m sufficiently divisible (independent of t).



Write Δ∼_T,Δ'+A
where A is ample over T
and the pair (X,Δ') is terminal.
By shrinking T, we may assume that no fibers of f are contained in Δ, Δ', or A.
Let ψ X'→ X be a log resolution
of (X, supp(Δ)+ supp(B)). By the assumption on the strata of (X,B), after shrinking T, we may assume that T = 0 and that the following hold for every closed point t∈ T:

    
  * X'_t→ X_t
is a log resolution of
(X_t,  supp(Δ|_X_t)+ supp(B|_X_t)), and 
    
  * every dlt modification
(Y,B_Y)→ (X,B) restricts
to a dlt modification 
(Y_t,B_Y_t)→ (X_t,B_t). 


Now let π (Y, B_Y) → (X,B) be a dlt modification,
and let D be a Cartier divisor on Y. We may write D∼_π^*(D_X)+E_Y
where D_X is a -Cartier divisor on X
and E_Y is π-exceptional.
The log canonical places of (X,B) are horizontal over T by (<ref>), so
after replacing X' by a further blow-up along log canonical
places of (X,B), we obtain a commutative diagram as follows, where
ϕ is a birational contraction and ψ still satisfies properties (<ref>) and (<ref>) (without further shrinking T):

    (Y,B_Y) [d]_-π   
     
    (X,B) [d]     X' [ld][l]^-ψ@–>[ul]_-ϕ
     
    T.


Fix a rational number ϵ>0 small enough so that all π-exceptional divisors are non-canonical places of (X,Γ_ϵ (1-ϵ)B+ϵΔ').
Write

    ψ^*(K_X+Γ_ϵ) + F_ϵ = K_X'+D'_ϵ

where D'_ϵ and F_ϵ are effective snc divisors 
without common components, and F_ϵ is ψ-exceptional.
Since every π-exceptional divisor
is a non-canonical place of (X,Γ_ϵ),
we may assume that F_ϵ is ϕ-exceptional.
Note that (X', D'_ϵ) is a klt pair by construction.

By Lemma <ref>,
every exceptional non-terminal place of (X,Γ_ϵ) is a log canonical place of (X,B).
Therefore, every non-terminal center of (X,Γ_ϵ) is a stratum of 
(X,⌈ D'_ϵ⌉).
By the assumption on the strata of (X,B), every proper stratum of (X',⌈ D'_ϵ⌉) is horizontal over T and restricts to a proper stratum on each closed fiber. 
Therefore, by successively performing blow-ups 
of the strata of (X',⌈ D'_ϵ⌉), we may assume that (X',D'_ϵ) is a terminal pair (see, e.g., <cit.>).

Let δ>0 be a rational number small enough so that 
A_ϵ,δϵ A +δ D_X
is ample
and (X', D'_ϵ,δ D'_ϵ +δϕ^* E_Y) is terminal.
By <cit.>, for every sufficiently divisible m and for every closed point t∈ U, the homomorphism

    H^0(X',𝒪_X'(m(K_X'+D'_ϵ,δ
    +ψ^*A_ϵ,δ
    )))
    → 
    H^0(X'_t, 
    𝒪_X'_t(m(K_X'+D'_ϵ,δ
    +ψ^* A_ϵ,δ)|_X'_t))
 
is surjective.
Since ϕ is a birational contraction, we have π^* = ϕ_* ∘ψ^*, so

    ϕ_*(m(K_X'+D'_ϵ,δ+ψ^* A_ϵ,δ))∼_δ m D.

Now let m be divisible enough, and let 

    Γ_t ∈ H^0(Y_t,𝒪_Y_t(δ m D|_Y_t))
     = H^0(Y_t, 𝒪_Y_t(m(π^*(K_X+Γ_ϵ + A_ϵ,δ) + δ E_Y)|_Y_t)) .

Pulling back Γ_t via ϕ_t  X'_t  Y_t, we obtain

    ϕ_t^*Γ_t ∈ H^0(X'_t, 𝒪_X'_t( m (K_X'+D'_ϵ,δ+ψ^* A_ϵ,δ-F_ϵ)|_X'_t)),

so we get

    ϕ_t^*Γ_t +mF_ϵ|_X'_t∈ 
    H^0(X'_t, 𝒪_X'_t( m (K_X'+D'_ϵ,δ+ψ^* A_ϵ,δ)|_X'_t)).

By the surjectivity of (<ref>), the section 
ϕ_t^*Γ_t+mF_ϵ|_X'_t is the restriction of a section

    Γ_X'∈ H^0(X',𝒪_X'(m(K_X'+D'_ϵ,δ+ψ^* A_ϵ,δ))).

By (<ref>), we conclude
ϕ_*(Γ_X')∈ H^0(Y,𝒪_Y(δ mD))
restricts to Γ∈ H^0(Y_t,𝒪_Y_t(δ mD|_Y_t)).



Let f X→ T be a fibration.
We say that f satisfies the extension
of prime divisors property if for every closed fiber X_t
and every effective prime divisor P_t⊂ X_t,
there an effective divisor P⊂ X 
for which supp(P)|_X_t= supp(P_t).



Next, we show that if a log CY fibration satisfies certain nice properties,
then after taking an étale cover of the base,
we can find a nice dlt modification with the property that any higher dlt modification has the extension of prime divisors property.
The proof of the following proposition uses a result of Totaro <cit.> on deformation invariance of the divisor class group for certain families of Fano varieties, after an étale cover of the base.


Let X be a terminal variety,
f X→ T a Fano fibration,
and (X,B) an lc pair that is log CY of index one over T.
Assume that every fiber of f is -factorial.
After replacing T by a dense open subset,
there exists a commutative diagram:

    (X,B)[d]_-f    (X_U,B_U)[d]^-f_U[l]_-ϕ    (Z,B_Z) [l] [dl]^-f_Z
    
    T     U[l]

satisfying the following conditions:


  * U→ T is a dominant finite étale morphism, the pair (X_U,B_U) is obtained by base change, and B_U contains no fibers of f_U, 

  * (Z,B_Z)→ (X_U,B_U) is a -factorial dlt modification that induces -factorial dlt modifications on log fibers over closed points,



  * the fibration f_Z is of Fano type, 

  * there is a boundary Δ_Z on Z such that
(Z,Δ_Z) is terminal and log CY over U, Δ_Z contains no fibers of f_Z, and all log fibers of (Z,Δ_Z)→ U are terminal, 

  * every stratum of (Z,B_Z) restricts to a stratum on each closed log fiber of f_Z, 

  * every -factorial dlt modification ϕ_Y (Y,B_Y)→ (Z,B_Z)
induces a Q-factorial
dlt modification on every closed fiber of (Z,B_Z)→ U, and 

  * for any -factorial dlt modification ϕ_Y (Y,B_Y)→ (Z,B_Z), the fibration f_Z∘ϕ_Y has isomorphic class group restrictions and satisfies the extension of prime divisors property. 




As X is terminal and is Fano over T,
then after possibly shrinking T, we can find a boundary Γ such that (X,Γ) is terminal and log Calabi–Yau over T.
After further shrinking T, we may assume that neither Γ nor B contains any fiber of X→ T, and that the fibers of f are Fano varieties with terminal singularities.

We first construct U→ T and (X_U,B_U) as in (<ref>).
Since each fiber of X→ T is -factorial and terminal, it has rational singularities <cit.>
and is smooth in codimension 2.
Moreover, by Kawamata–Viehweg vanishing, every fiber has acyclic structure sheaf,
so we may apply <cit.> to find a dominant finite étale morphism T'→ T from a smooth variety 
such that the base change f' X'  X_T'→ T' has the property
that Cl(X'/T') maps surjectively to the class group of each closed fiber.
Let (X',B') (resp. (X',Γ')) be 
the log pull-back of (X,B) (resp. (X,Γ)) to X'.
Note that f' is still a Fano type fibration,
(X',B') is log Calabi–Yau of index one over T',
and (X',Γ') is terminal and log Calabi–Yau over T'.
By <cit.> applied to any dlt modification of (X',B'), we can find a dominant finite étale morphism
U→ T' from a smooth variety
such that if f_U (X_U,B_U)→ T”' 
is the base change,
then the dual complex of (X_U,B_U)
restricts to the dual complex of the log fibers. 
In particular, each prime component of B_U 
remains prime after restriction to any closed fiber.
We let (X_U,Γ_U) be the log pullback of (X',Γ') to X_U.
The morphism f_U is Fano type, 
(X_U,B_U) is log CY of index one over U,
and (X_U,Γ_U) is terminal and log CY over U.
Further, (X_U/U) maps surjectively to the class group of each closed fiber of f_U.

Now we turn to constructing the dlt modification
(Z,B_Z)→ (X_U,B_U) in (<ref>).
By Lemma <ref>, 
for sufficiently small ϵ>0
we can find a -factorial dlt modification
(Z,B_Z)→ (X_U,B_U)
such that
the log pullback (Z,Δ_Z)
of (X_U,(1-ϵ)B_U+ϵΓ_U)
is terminal.
We argue that the closed fibers of Z→ U remain -factorial.
For a closed point t∈ U, the prime exceptional divisors of Z_t→ X'_t
are restrictions of the prime exceptional divisors of Z→ X'.
Indeed, this holds as the prime exceptional divisors of Z→ X' are components of B_Z and hence correspond to vertices of the dual complex 𝒟(Z,B_Z).
The prime exceptional divisors 
of Z→ X' are -Cartier.
Thus, the exceptional prime divisors
of Z_t→ X'_t are -Cartier, and so Z_t is -factorial.
We conclude that the -factorial 
dlt modification (Z,B_Z)→ (X_U,B_U) induces -factorial dlt modifications on log fibers over closed points. This proves (<ref>).
Moreover, as the exceptional prime divisors over closed fibers are restrictions of the exceptional prime divisors of Z → X_U, the composition f_Z  Z → U has surjective class group restrictions.
We have two pairs on the Q-factorial variety Z, namely:

    
  * (Z,B_Z), which is dlt and log CY of index one over U, and
    
  * (Z,Δ_Z), which is terminal and log CY over U.

The morphism f_Z is Fano type by Lemma <ref>. 
After possibly shrinking U again, we may assume that U is smooth and affine,
all the log fibers of (Z, Δ_Z)→ U are terminal,
and f_Z has isomorphic class group restrictions (by Lemma <ref>(<ref>)).
The following conditions are satisfied:


    
  * the pair (Z, B_Z) is -factorial,
    dlt, and log CY of index one over U, 
    
  * the morphism f_Z is of Fano type and has isomorphic class group restrictions, 
    
  * the pair (Z,Δ_Z) is terminal
    and log CY over U, 
    
  * each stratum of (Z,B_Z) restricts to a stratum of the log fiber over each closed point,
    

showing that the assumptions of Lemma <ref> hold, so after shrinking U again, we furthermore have

  
  * for every t∈ U, every -factorial dlt modification ϕ_Y (Y,B_Y) → (Z,B_Z),
    and every divisor D on Y, we have a surjective homomorphism

    H^0(Y,𝒪_Y(mD)) ↠ H^0(Y_t,𝒪_Y_t(mD|_Y_t)),

for every m sufficiently divisible. 


Now we verify the conditions in the proposition statement. We've already seen that (<ref>) and (<ref>).
Conditions <ref> and <ref> imply (<ref>) and (<ref>), respectively.
Conditions <ref> and <ref>
ensure that (<ref>) and (<ref>) hold.
Thus, it remains to show (<ref>).

First, we argue that f_Z∘ϕ_Y has isomorphic class group restrictions.
For surjectivity, the class group of every fiber Y_t is generated by the pullbacks of divisors from Z_t
and the prime exceptional divisors of Y_t→ Z_t.
By <ref> and <ref>,
the prime exceptional divisors of Y_t→ Z_t are restrictions of the prime exceptional divisors of Y→ Z.
Since Y is Q-factorial, this shows Y_t is also Q-factorial.
For injectivity,
if D_t is a -Cartier divisor on Y that is -linearly trivial on Y_t,
then ϕ_Y,t_*D_t is -linearly trivial on Z_t.
The pushforward ϕ_Y_*D is -linearly trivial over U by <ref>,
so D ∼_U, E for some ϕ_Y-exceptional divisor E.
Since E_t is -linearly trivial on  Y_t and exceptional over Z_t, the negativity lemma implies that E_t is the trivial divisor.
Hence, since E is horizontal over the base by <ref>, we conclude that E is trivial, so D is -linearly trivial over U.

It remains to show that f_Z∘ϕ_Y satisfies the extension of prime divisors property.
Let E_t⊂ Y_t be an effective prime divisor.
Since f_Z∘ϕ_Y has isomorphic class group restrictions,
there is a divisor D on Y for which D|_Y_t∼_ Q E_t,
or, in other words, 0≤ mE_t ∈ H^0(Y_t,𝒪_Y_t(mD|_Y_t)) for some m.
By condition <ref>, after replacing m by a sufficiently divisible multiple, there exists 
0≤ F ∈ H^0(Y,𝒪_Y(mD)) 
such that F|_Y_t=mE_t. 
In particular, F is an effective divisor
for which supp(F)|_Y_t= supp(E_t).



Let X be a terminal variety, 
f X→ T a Fano fibration,
and (X,B) an lc pair
that is log CY of index one over T.
Assume that every fiber of f is -factorial.
There is a non-empty open subset V⊂ T such that if (X_t,B_t) is of cluster type for some t∈ V, then there exists an open neighborhood t∈ V_0 ⊆ V over which every fiber is of cluster type.



Consider the commutative diagram
provided by Proposition <ref>:

    (X,B)[d]_-f    (X_U,B_U)[d]^-f_U[l]_-ϕ    (Z,B_Z) [l] [dl]^-f_Z
    
    T     U.[l]

By Proposition <ref>(<ref>) there exists a boundary Δ_Z such that (Z,Δ_Z) is terminal and log Calabi–Yau over U and Δ_Z contains no fibers of f_Z.
Furthermore, (Z,B_Z)→ (X_U,B_U) induces -factorial dlt modifications on closed fibers by Proposition <ref>(<ref>).
After possibly shrinking U, we may assume that the following conditions are satisfied:

    
  * U is a smooth affine variety, 
    
  * f_Z has equidimensional fibers, 
    
  * for every ϵ∈ [0,1], the boundary divisor and the moduli divisor induced by the canonical bundle formula by (Z,(1-ϵ)B_Z+ϵΔ_Z) on U are trivial, and 
    
  * no fiber of Z → U is contained in B_Z or Δ_Z, every log fiber of (Z,B_Z)→ U is dlt,
    and every log fiber of (Z,Δ_Z)→ U is terminal. 
    

Let V ⊂ T be an open subset contained in the image of U;
we may further assume V is affine.
We will show that V is the desired open subset in the proposition statement.

Thus, assume that (X_t_0,B_t_0) is of cluster type 
for some t_0∈ V,
and let s_0∈ U be a pre-image of t_0.
Then the dlt log CY pair (Z_s_0,B_Z,s_0)
is of cluster type by Lemma <ref> and Proposition <ref>.
By Lemma <ref>, 
we have a commutative diagram

    (Y_s_0,B_Y,s_0)[d]_-ϕ_Y,s_0@–>[rd]^-π_s_0   
    
    (Z_s_0,B_Z,s_0) @–>[r]     (^d,Σ^d),

where ϕ_Y,s_0 (Y_s_0,B_Y,s_0)→ (Z_s_0,B_Z,s_0)
is a Q-factorial dlt modification
and 
π_s_0 (Y_s_0,B_Y,s_0)  (^d,Σ^d)
is a crepant birational contraction.
Furthermore, if E_s_0 Ex(π_s_0)∖ B_Y,s_0,
then (Y_s_0,B_Y,s_0+ϵ E_s_0)
is dlt for ϵ>0 small enough.
By Lemma <ref>,
we may pass to a higher dlt modification to assume that ϕ_Y,s_0 is
obtained by
blow-ups of strata of (Z_s_0,B_Z,s_0).
The strata of (Z_s_0,B_Z,s_0)
are precisely the restrictions of the strata
of (Z,B_Z) to the log fiber over s_0
by condition (<ref>).
Hence, there exists a Q-factorial dlt modification (Y,B_Y)→ (Z,B_Z),
obtained by a sequence of
blow-ups of strata of (Z,B_Z),
such that the log fiber over s_0
of (Y,B_Y)→ U is precisely
(Y_s_0,B_Y,s_0).

We will show that, over a neighborhood U_0 of s_0, the log fibers of (Y, B_Y) are cluster type CY pairs. To do this, we will construct a birational model Y  W on which we may apply Proposition <ref>.
We show that Y W induces
fiberwise birational contractions to toric log CY pairs. We construct W as follows.

By Proposition <ref>(<ref>), there exists an effective divisor E_Y on Y such that supp(E|_Y_s_0)= supp(E_s_0). Note that E_Y has no common components with B_Y.
By Lemma <ref>, after possibly replacing (Y,B_Y) with a higher dlt modification,
we may assume that (Y,B_Y+δ E_Y) 
is dlt for δ>0 small enough.
In particular, over an open neighborhood s_0 ∈ U_0 ⊂ U, the effective divisor E_Y contains no fiber of Y_U_0→ U_0 and no intersection of a log canonical center of (Y,B_Y) with a fiber of Y_U_0→ U_0.
By Lemma <ref>, Lemma <ref>, and Proposition <ref>(<ref>), after possibly replacing  (Y,B_Y)
with a higher -factorial dlt modification, we may assume that for ϵ > 0 small enough, the log pull-back 
(Y,Γ_Y) of (Z,(1-ϵ)B_Z+ϵΔ_Z) to Y
is terminal and the log fibers of (Y,Γ_Y) → U are terminal.
Then, for δ>0 small enough, the pair
(Y,Γ_Y+δ E_Y) is terminal
and its restriction to closed fibers of Y_U_0→ U_0 are terminal.
By Lemma <ref>, the morphism  f_Z∘ϕ_Y is Fano type.
Hence, we may run a (K_Y+Γ_Y+δ E_Y)-MMP over U_0
that terminates with a good minimal model W→ U_0.

Let B_W (resp. E_W) be the pushforward of B_Y, U_0 (resp. E_Y, U_0) to W.
Next, we will show that (W, B_W) satisfies the hypotheses of Proposition <ref> over some open neighborhood of s_0.

First, W → U_0 has injective class group restrictions by Proposition <ref>(<ref>) and Lemma <ref>.
Next, we apply Lemma <ref> to (Y,Γ_Y)
and Lemma <ref> to (Y,B_Y) over U_0.
The assumptions on Lemma <ref> hold by <ref>, the choice of ϵ above, and <ref>.
The assumptions on Lemma <ref> hold by <ref>, Proposition <ref>(<ref>), and <ref>.
By Lemma <ref>(<ref>) and (<ref>), we conclude that W_s is normal for every closed s∈ U,
and the corresponding birational maps Y_s W_s
are birational contractions. In particular Y_U_0 W is a birational contraction; furthermore, by <ref>, W → U_0 has equidimensional fibers.
By Lemma <ref>, the pair (W_s,B_W, s) is log canonical for every s∈ U_0.

Next, since Y_U_0 W is a birational contraction and contracts precisely E_Y, U_0, the negativity lemma implies that (W, B_W) is a log CY pair and (Y_U_0, B_Y, U_0)  (W, B_W) is crepant birational. Furthermore, over each closed point s∈ U_0, the same argument and conclusions hold for the log fibers.
Over the point s_0,
since E_s_0 is π_s_0-exceptional,
the composition (W_s_0,B_W,s_0) (Y_s_0,B_Y,s_0)  (^d,Σ^d) is a crepant birational contraction that only contracts log canonical places of (^d,Σ^d).
So (W_s_0,B_W,s_0) is a toric log CY pair by Lemma <ref>.
Thus, we have a Q-factorial pair (W,B_W) that is log CY
of index one and satisfies the following conditions:

    
  * W_s =  W_s_0 for every s∈ U_0,
    
  * the divisor B_W is reduced and contains no fibers of W→ U_0,
    
  * the log fibers of (W,B_W)→ U_0 are log canonical pairs,
    
  * the restriction of B_W to B_W,s_0 induces a bijection between prime components, 
    
  * the fibration W → U_0 has injective class group restrictions, and
    
  * (W_s_0,B_W,s_0) is a toric log CY pair.

By Proposition <ref>, all fibers of (W,B_W)→ U_0 are toric log CY pairs.

Let t_0 ∈ V_0 ⊂ V be an open subset whose preimage in U is contained in U_0. We will show that the log fiber (X_t, B_t) is a cluster type log CY pair for any closed point t∈ V_0.
If s∈ U_0 is a preimage of t,
then we have a Q-factorial dlt modification
(Y_s,B_Y,s)→ (Z_s, B_Z,s) → (X_s, B_s)
by Proposition <ref>(<ref>) and (<ref>).
By construction of U_0, this admits a crepant birational contraction (Y_s,B_Y,s) (W_s,B_W,s) to a toric log CY pair. Thus, Lemma <ref> shows that (X_s, B_s) ≅ (X_t, B_t) is of cluster type.



We proceed by Noetherian induction. 
If T=0, then there is nothing to prove, so assume T≥ 1.
Take a dominant finite étale morphism T' → T such that the base change (𝒳_T', ℬ_T') is a log CY pair of index one over T'.
Let V'⊆ T' be the open subset 
provided by Theorem <ref>,
and let V ⊆ T be its image.
Then T'_ cluster∩ V'
is a constructible subset by Theorem <ref>, and Noetherian induction,
so its image T_ cluster∩ V in T is also constructible.
By Noetherian induction, we conclude that T_ cluster is a constructible subset of T.



The terminal and Q-factorial assumptions on the fibers are necessary for our proof of Theorem <ref>, in particular for our method of spreading out the crepant birational contraction from (Y_s_0, B_Y, s_0) to a toric log CY pair in the proof of Theorem <ref>.
We use the terminality assumption to apply Lemma <ref>. We use the terminality and Q-factoriality assumptions to apply <cit.> in the proof of Proposition <ref>.





§ PROOFS OF MAIN THEOREMS

Now we turn to proving the two main theorems of this article, using Theorems <ref> and <ref>. We will need the following lemmas.


Let (𝒳,ℬ)→ T be a family of log CY pairs.
Then there is a locally closed stratification of T such that the log fibers over each stratum have the same index.



By Noetherian induction, it suffices to show that there exists a non-empty open set of T over which all log fibers have the same index.
As the geometric generic log fiber is log CY, we may find a dominant finite étale morphism
T'→ T such that
Pic(T')=0 and
the base change
(𝒳_T',ℬ_T') → T' is a log CY fibration.
Thus, over a non-empty open subset U' ⊂ T', we may find the smallest positive integer m for which m(K_𝒳_T'+ℬ_T')|_𝒳_U' = m(K_𝒳_U'+ℬ_U')∼ 0.
In particular, the index of each log fiber of (𝒳_U',ℬ_U')→U' divides m.
Let 𝒴_U'→𝒳_U' be the index one cover of K_𝒳_U'+ℬ_U'.
By construction, the log fibers of (𝒳_U',ℬ_U')→U' with index m are precisely the connected fibers of 𝒴→U'.
Hence, there is an open subset of U' over which all log fibers of (𝒳_U',ℬ_U')→U' is exactly m.



Let (𝒳,ℬ)→ T be a family of pairs.
Then the subset of T parametrizing log fibers with log canonical singularities is constructible. 



Let T'→ T be a dominant finite étale morphism such that the base change (𝒳_T', ℬ_T') admits a log resolution 𝒳' →𝒳_T' over T' and such that ℬ_T' does not contain any fibers of 𝒳_T'→ T'.
By Noetherian induction, it suffices to show that the condition holds over some non-empty open subset of the base.
There is a non-empty open subset U⊂ T' over which all the exceptional divisors of 𝒳' are horizontal over U.
By possibly shrinking U, we may assume that 𝒳'→𝒳_T' induces a log resolution on log fibers of (𝒳_T',ℬ_T')→ T over closed points of U.
Therefore, over U, either all or none of the log fibers of (𝒳,ℬ)→ T are log canonical.
This finishes the proof.



By Noetherian induction,
it suffices to show that T_ toric∩ U is constructible for some open dense affine subset U⊆ T.
By <cit.>, there is a dense open subset U⊂ T such that for m divisible enough, the map
H^0(𝒳,-mK_𝒳)→
H^0(𝒳_t,-mK_𝒳_t)
is surjective (and nonzero) for every closed point t∈ U.
Fix such an m,
let 

    ℬ⊂𝒳_U× |-m K_𝒳_U|

be the universal divisor, and consider the family π (𝒳_U × |-m K_𝒳_U|, 1/mℬ) → U × |-m K_𝒳_U|.
The fiber of π over (t,p) is (𝒳_t, 1/m B_p|_𝒳_t), where ℬ_p ∈ |-m K_𝒳| is the effective divisor corresponding to p.
Let W⊆ U × |-m K_𝒳_U| be the open subset consisting of points (t,p) for which B_p does not contain the fiber 𝒳_t, and write (𝒴_W,ℬ_W) → W for the restriction of the family π to W.

Then (𝒴_W,ℬ_W) → W is a family of pairs.
Let W_ toric⊂ W be the locus parametrizing the log fibers that are toric log CY pairs.
Since a variety X is toric if and only if it admits a boundary B for which (X,B) is a toric log CY pair,
the intersection T_ toric∩ U is equal to the image of W_ toric under the projection W ⊂ U × |-mK_𝒳_U|  U. Thus, it suffices to show that W_ toric⊂ W is a constructible subset.

By Lemma <ref>, W_ toric is contained in the set W_1 parametrizing log canonical log fibers of index one, and W_1 ⊂ W is constructible
by Lemmas <ref> and <ref>.
Over the normalization of each irreducible component of W_1, the pair
(𝒴_W,ℬ_W) induces a family of log CY pairs of index one for which every closed fiber 𝒴_t is Fano.
By Theorem <ref>, we conclude that the subset 
W_ toric⊆ W_1 ⊆ W
parametrizing toric log CY fibers is constructible,
as it is a constructible subset of a constructible subset. This completes the proof.



The proof is identical to the proof of Theorem <ref>, using Theorem <ref> instead of Theorem <ref>.



The n-dimensional smooth Fano varieties form a bounded family by <cit.>; that is, there exist finitely many surjective projective morphisms g_i𝒵_i → S_i of quasi-projective varieties such that any n-dimensional smooth Fano variety is isomorphic to a closed fiber of g_i for some i (see, e.g., <cit.>).
Furthermore, by standard arguments, one may replace each S_i by a constructible subset to assume every closed fiber is an n-dimensional smooth Fano variety, i.e., that the g_i are parametrizing families for n-dimensional smooth Fano varieties. Then the corollary is immediate from Theorem <ref>.




§ EXAMPLES AND QUESTIONS


In this section, we collect some examples
of the behavior of toricity and cluster type
in families of Fano varieties.
We propose some questions for further research.
The first two exmaples show that the property of being a toric variety is neither open nor closed. 


Many smooth Fano varieties admit singular toric degenerations.
For instance, the Mukai–Umemura threefold V_22 admits a degeneration to a Gorenstein terminal toric Fano 3-fold (see, e.g., <cit.>). In this degeneration, all the fibers except the central fiber are non-toric.



[<cit.>]
The projective plane ℙ^2 admits non-toric, quasismooth degenerations.
Namely, let (a,b,c) and (a,b,d) be two pairs of Markov triples for which 
d=3ab-c, the so-called adjacent Markov triples.
Consider the weighted hypersurface

    V(x_1x_2+x_3^c+x_4^d)⊂(a^2,b^2,d,c) .

By <cit.>, this 
is a non-toric, quasismooth, rational, Fano 𝔾_m-surface of
Picard rank one,
which is a degeneration of ^2.
Furthermore, by <cit.>, all the non-toric,
normal, rational projective
𝔾_m-surfaces which are degenerations of ^2 arise in this way.
Hence, the property of being a toric variety is not closed in families of Fano varieties.



The following example shows that the property of being cluster type
is not open in families of Fano varieties.
We give a family of canonical Fano threefolds such that the special fiber is of cluster type but not toric, and nearby fibers are irrational and hence not of cluster type.


Let X ⊂ P^4 be the singular cubic threefold defined by

    f(x_0,…,x_4)  (x_0 + x_1 + x_2 + x_3 + x_4)^3 - x_0^3 - x_1^3 - x_2^3 - x_3^3 - x_4^3.

Let g(x_0,…,x_4) be the equation of a general cubic, and
consider the family of Fano threefolds

    𝒳{ ([x_0:…:x_4],t) |  (1-t)f(x_0,…,x_4)+tg(x_0,…,x_4)=0}⊂^4 ×𝔸^1
 
over 𝔸^1.
The general fiber of 𝒳→𝔸^1 is a smooth cubic threefold, so it is irrational.

Now we show that the special fiber 𝒳_0 = X is of cluster type but not toric. Since X is isomorphic to the Segre cubic

    { [x_0:…:x_5] | x_0+…+x_5 = x_0^3+…+x_5^3=0}⊂ℙ^5,

it has 10 nodal singularities and
its automorphism group is S_6; in particular, X is not a toric variety.
It remains to construct a boundary B on X such that (X,B) is a log CY pair of cluster type.
There is a small resolution π Y→ X
which is isomorphic to the blow-up ϕ Y →^3
at 5 general points
{p,q,r,s,t}⊂ P^3 <cit.>.
Let H_1 (resp. H_2) by the hyperplane in ^3
spanned by {p,q,r} (resp. {p,s,t}).
Since {p,q,r,s,t} are in general position, the line ℓ_1  H_1 ∩ H_2 only contains one point of the set {p,q,r,s,t}, namely p. 
Next, let u_1 and u_2 be two general points in ℓ_1,
and let H_3 (resp. H_4) be the hyperplane in ^3 spanned by {u_1,q,s}
(resp. {u_2,r,t}).
By construction, the pair
(^3,H_1+H_2+H_3+H_4) is log Calabai–Yau,
and each point of the set {p,q,r,s,t} is contained in precisely two of the hyperplanes
{H_1,H_2,H_3,H_4}. 
In particular, if we write 

    ϕ^*(K_^3+H_1+H_2+H_3+H_4)=K_Y+B_Y,

then the divisor B_Y is effective. 
Then (X,B π_* B_Y) is a log CY pair,
and the exceptional set of 
the crepant birational map

    π∘ϕ^-1 (^3,H_1+H_2+H_3+H_4)  (X,B)

is precisely the 10 lines passing through pairs of points in the set {p,q,r,s,t}.
In particular, the intersection 𝔾_m^3∩ Ex(π∘ϕ^-1) contains no divisors, 
so the pair (X,B) is of cluster type.
This shows that the Fano variety X is of cluster type.



The following example shows that the property of being cluster type is not closed in families of Gorenstein canonical Fano surfaces. 


Let X_0 be the Gorenstein del Pezzo surface of Picard rank one with a single D_5 singularity.
By <cit.>, we know that X_0 admits a -Gorenstein smoothing 𝒳→𝔸^1.
Since the anticanonical volume of X_0 is 4 (see, e.g., <cit.>),
the general fiber of 𝒳→𝔸^1 is a smooth del Pezzo surface of degree 5.
Every smooth del Pezzo surface of degree 5
admits a 1-complement of coregularity zero <cit.>
and so it is of cluster type by <cit.>. 
On the other hand, the central fiber 𝒳_0 is not of cluster type, 
as cluster type surfaces only admit A-type singularities (see <cit.>).



The following example shows that the property of being a (possibly non-normal) toric variety is not locally closed in families of slc varieties.


Consider the trivial family π_2𝒴^2×𝔸^1_t→𝔸^1_t.
Now, let 𝒟 be a divisor on 𝒴 for which 𝒟_0 is the reduced sum of the coordinate hyperplanes
and 𝒟_t is a smooth cubic curve for every t≠ 0. 
Let 𝒳 be the deformation of 𝒴 to the normal cone of 𝒟.
For each t∈𝔸^1, we let ψ_t 𝒳_t →𝔸^1_s
be the induced deformation of 𝒴_t to the normal cone over 𝒟_t.
Then, the threefold 𝒳 admits a morphism
ψ𝒳→𝔸^2 whose fiber over (s,t) is ψ_t^-1(s).
By construction, all the fibers of ψ over 𝔾_m×𝔸^1_t are isomorphic to ^2 and hence toric varieties.
The fibers over {0}×𝔾_m are cones over smooth cubic curves and so they are not toric. 
The fiber over (0,0) is the non-normal cone over the reduced sum of the hyperplane coordinates of ^2, which is a toric variety.
We conclude that the subset of 𝔸^2
parametrizing toric fibers of ψ𝒳→𝔸^2 is

    (𝔾_m×𝔸^1) ∪ (0,0)

which is not a locally closed subset. However, it is a constructible subset.



The previous examples motivate the following question.


    Is log rationality a constructible condition in families of Fano varieties?


Rationality specializes in smooth families <cit.>. 
Thus, the following question seems to be reasonable
to study log rationality in families.


 Let (𝒳,ℬ)→ C be a log smooth family for which the generic fiber is log rational. Are closed fibers log rational?


The previous question is closely related to the log rationality conjecture proposed
by Ducat and Enwright, Figueroa, and the second author (see <cit.> and <cit.>).

Finally, our proof of Theorem <ref> requires that the fibers of the family  are Q-factorial and terminal (see Remark <ref>). However, we are not aware of any counterexamples to Theorem <ref> when these assumptions on the singularities are removed.

    Is the property of being cluster type a constructible condition in families of Fano varieties with worse than terminal Q-factorial singularities?


habbvr


