Families of stable 3-folds in positive characteristic

. We show that ﬂat families of stable 3-folds do not lead to proper moduli spaces in any characteristic p > 0 . As a byproduct, we obtain log canonical 4-fold pairs, whose log canonical centers are not weakly normal.

Following [KSB88], the moduli space of varieties of general type is compactified as follows.Start with a family of smooth projective varieties g • : X • B → B • over a smooth, affine curve B • .Possibly after a base change C • → B • , it extends to a semi-stable family g : X → C over a smooth proper curve C ⊃ C • , as in [KKM + 73].Finally, let g c : X c → C be the relative canonical model as in [KM98,Section 3.8].
In characteristic 0, the resulting fibers have semi-log-canonical singularities and ample canonical class.These are called stable varieties.Stable varieties with a fixed volume have a coarse moduli space that is projective over Q; see [Kol23] for details.
Our aim is to give examples to show that this process does not work for 3-folds in positive characteristic.For simplicity, we work over algebraically closed fields from now on.
Theorem 1. Flat families of stable 3-fold pairs of fixed volume do not form a proper moduli theory in any characteristic p > 0.
Remark 2. In our examples, the approach of [KSB88] does produce a canonical model g c : X c → C; the problem is that some of its fibers do not satisfy Serre's condition S 2 .However, the normalization of each fiber is a stable variety.So we may well have a proper moduli space, but we need to allow some families whose fibers are stable only after normalization.However, we do not have a precise conjecture on which families should be allowed.
It is known that moduli theory is more complicated in positive characteristic.This is partly due to the failure of Kodaira's vanishing theorem, and to the current lack of resolution of singularities.The appearance of p-torsion in the class group leads to other problems.However, it was expected that once these are correctly accounted for, the rest of the arguments would go through.For surfaces, large parts of the theory have been worked out in [Pat17,ABP23].
Surprisingly, in Example 4 the normalization of the central fiber does lift to characteristic 0, so Kodaira's vanishing theorem holds on it by [DI87].The problem comes from the nearby fibers that exist only in characteristic p > 0. These nearby fibers are normal, but not CM by Lemma 17.In characteristic 0, being CM is a deformation-invariant property of stable varieties; see [KK10].It is thus possible that such examples do not occur if one works solely in the closure of the moduli of smooth varieties of general type in dimensions 2 and 3; see [ABL22,Bri22].As we discuss in Example 3, there are such examples starting in dimension 5 in characteristic 2.
Therefore, while the examples obtained so far indicate that the positive-characteristic moduli theory is much more subtle, they do not rule out the possibility that, in the end, the necessary modifications are mainly technical.
A series of non-CM singularities is discussed in the papers [Kov18, Tot19, Ber19], but it did not seem to have been observed that they can be used to construct stable degenerations, where the generic fibers are smooth with ample canonical class, and the special fibers have isolated, non-normal singularities.The dimension of the resulting examples is about twice the characteristic.
Example 3. (Kovács-Totaro-Bernasconi examples).Let X = G/P be a projective, homogeneous space.The cases when P is non-reduced were studied in [HL93,Lau96,LR97].Most of these are not Fano, but if X is Fano and Kodaira vanishing fails, then cones over X give interesting singularities; see [Kov18,Tot19,Ber19].The smallest dimension of such an X is 5 in characteristic 2, and about twice the characteristic in general.
Assume that −K X = mH for some ample divisor H for some m ≥ 1.Then |H| is very ample by [Lau96], so it gives an embedding X → P N , where N = dim |H|.Let Y := C(X, H) ⊂ P N +1 be the projective cone over X with vertex v. Then There is a natural morphism π : C(D, H| D ) → D Y , which is an isomorphism outside the vertex.Furthermore, π is an isomorphism if and only if depth v D Y ≥ 2, which holds if and only if depth v Y ≥ 3. Thus, if H 1 X, O X (H) 0, then D Y is not normal.Intersecting Y with a pencil of hyperplanes gives a locally stable degeneration with generic fiber X and special fiber D Y .Taking a suitable cyclic cover as in [KM98, Section 2.4], we get a series of examples of stable degenerations, where the generic fibers are smooth varieties with ample canonical class, and the special fibers have isolated non-normal singularities.
The cases described in [Tot19] have m = 2. Then the normalization of D Y has canonical singularities; hence these deformations take place in what is usually considered the 'interior' of the moduli space.On the other hand, these constructions start with a variety for which Kodaira vanishing fails, so they cannot be lifted to characteristic 0.
Aside.The series of non-CM, quotient singularities of [Yas19] all have depth ≥ 3 by [ES80], so they do not give similar examples.
All the ingredients going into the proof of Theorem 1 seem to be well known.Unipotent bundles on elliptic curves have been studied in [Ati57,Har71,Oda71].Closely related examples of pathological families of elliptic surfaces are given in [Bri20].Using cones to go from a lower-dimensional non-general type variety to a higher-dimensional general type one has been utilized many times.For example, [McK06] used it to show that log abundance for general type varieties in dimension n + 1 implies log abundance for all varieties in dimension n; see Section 11.Related results on the deformation invariance of plurigenera also appear in [EH21,BBS21].
The main step is the following; see [KM98, Definition 2.37] for the definition of divisorial log terminal, abbreviated as dlt.
Example 4. In every characteristic p > 0, there exist a morphism g : Y → A 1 and an effective Q-divisor ∆ on Y such that (4.a) g : Y → A 1 is smooth, projective, of relative dimension 3; (4.b) K Y + ∆ is g-semiample and g-big; (4.c) the fibers (Y t , ∆ t ) are dlt;

and this continues to hold after any base change
for t 0 and m ≥ 1 sufficiently divisible.As in [Kol13, Theorem 4.9], we have (4.d)⇒ (4.c) by adjunction.In characteristic 0, inversion of adjunction says that (4.c) implies the log canonical variant of (4.d).The key new feature is the jump of the plurigenera (4.e).
There are many examples in positive characteristic where finitely many of the plurigenera jump.The surfaces produced in [Bri20] lead to families of elliptic pairs (with terminal singularities) where infinitely many of the plurigenera jump.These surfaces can also be used (instead of Example 8) to obtain families of 3-folds as in Example 4.
The new feature of Example 4 is that the fibers are of general type, and infinitely many of the plurigenera jump.This has strong consequences.
Proof of Theorem 1.We deduce the theorem from Example 4. Since K Y + ∆ is g-semiample and g-big, there exist a morphism with connected fibers h : Y → Y and a relatively ample divisor On the right-hand side, we can change the summation to multiples of m 0 , which shows that Y c , ∆ c = Y , h * ∆ .The fiber over the origin is The properties (4.a)-(4.e)continue to hold after any base change is the unique stable degeneration of the family over In our examples, Sing(Y t ) c is 1-dimensional.Localizing at its generic point gives a simple elliptic singularity of dimension 2. There is a natural morphism which is an isomorphism outside the singular set and purely inseparable over Sing(Y c ) 0 .These imply the following; see [Kol96, Section 7.2] for weak normality.
Corollary 5.For any p > 0, there are log canonical 4-fold pairs (X, S + ∆) of characteristic p, where S is a Cartier divisor that is not weakly normal.
Proof.The pair Y c , ∆ c + (Y c ) 0 is a log canonical 4-fold, (Y c ) 0 is a Cartier divisor and a log canonical center but not weakly normal.
Remark 6.In characteristic 0, weakly normal coincides with seminormal, and all log canonical centers are seminormal by [Amb03,Fuj17]; see also [Kol13,Theorem 5.14].A series of papers culminating in [BMP + 20] establish that MMP for 3-folds works in characteristics at least 7, just as in characteristic 0. This led to a hope that, in any fixed dimension, new phenomena appear only in low characteristics; see [Tot19, Ber19, HW19, HW20] for such results.By Corollary 5, this is not the case for 4-folds.
However, in MMP one can frequently work with dlt pairs only; thus a dlt example would be the main test case.Our Y c , ∆ c are not dlt; see Lemma 17.
Let (X, S) be a 3-dimensional plt pair.If char ≥ 7, then S is normal, but there are counterexamples in characteristics 2; see [CT19].See also [Ber21,Lac20,ABL22] for related examples.However, no such example seems to be known where S is Cartier.Question 7. Let (X, S + ∆) be a 3-dimensional, semi-log-canonical pair, where S is a Cartier divisor.Is S weakly normal?
We start the construction of our 3-dimensional examples with a family of elliptic surfaces, similarly to [Bri20].Then we build these up to dimension 3.
Example 8. We construct a morphism g : (S, ∆ + Θ) → A 1 of relative dimension 2, where g : S → A 1 is smooth, projective, the fibers (S t , ∆ t + Θ t ) are terminal, K S + ∆ is g-semiample, numerically trivial, and K S + ∆ + Θ is g-semiample of Kodaira dimension 1.Moreover, all large plurigenera jump: The key is the behavior of unipotent vector bundles on elliptic curves in positive characteristic.9. Elliptic ruled surfaces.Let E be an elliptic curve.A vector bundle is unipotent if it can be written as a successive extension of copies of O E .
By [Ati57], for every r, there is a unique rank r, indecomposable, unipotent vector bundle F r = F r (E), and every unipotent vector bundle is a direct sum of these F r .Thus a unipotent vector bundle U is isomorphic to F r if and only if rank U = r and h 0 (E, U ) = 1.
Note that F 2 (E) sits in an exact sequence Let τ : E → E be a non-constant map of elliptic curves.Then we get This shows that ) is an injection.In particular, we are always in the first case in characteristic 0.
Note that τ * : is the zero map.(For higher-dimensional Abelian varieties, ρ * has a 1-dimensional kernel; see [Mum70, Section 15].) Projectivising F 2 , we get a P 1 -bundle π 1 : S 1 → E with a unique section D 1 with self-intersection 0. Note that K S 1 + 2D 1 ∼ 0 and the normal bundle of D 1 is trivial.
Let C ⊂ S 1 be an irreducible, reduced curve that is disjoint from D 1 .Then it is numerically equivalent to a multiple of D 1 ; hence p a (C) = 1 by adjunction.The projection τ : C → E is dominant; hence C is a smooth, elliptic curve.The fiber product S 1 × E C has two disjoint section; hence τ * F 2 (E) is trivial.Thus Conversely, let g : C → E be a non-constant morphism such that g * : Proof.Let C be an integral curve such that C ∼ mD 1 .Then (C • D 1 ) = 0, so either C and D 1 are equal, or they are disjoint.As we noted above, the latter cannot happen if either char k = 0 or m < char k.Thus h 0 (S 1 , O S 1 (mD 1 )) = 1 for such values.Pushing forward the filtration gives that (π 1 ) * O S 1 (mD 1 ) is unipotent.

Proof. The filtration
gives that H 0 (S 1 , O S 1 (mD 1 ) Note that if char k = 0, then in fact h 0 (S 1 , O S 1 (mD 1 )) = 1.If char k = p > 0, then we expect that h 0 (S 1 , O S 1 (mD 1 )) = m/p + 1. (See [Sch10] for computing F m ⊗ F n .) Remark 9.4.The reader can check that in our constructions, one can replace E with any Abelian variety A of dimension n.The key property is that there is a non-split extension that becomes split after a suitable degree p cover A → A. By [Mum70, Section 15], this always holds in characteristic p > 0. The resulting g : (Y , ∆) → A 1 has relative dimension n + 2. The singular sets of the canonical models (Y t ) c are still 1-dimensional.

Construction of Example 8.
Over A 1 , there is a degeneration of F 2 to the split bundle O E ⊕ O E .Thus we get a vector bundle Let π : S → E × A 1 be the corresponding P 1 -bundle with the unique section The jump in the plurigenera comes from Claim 9.3: It is nef, and by the above computations, all the plurigenera of the fibers jump at c = 0.However, (S t , 2D t ) is not log canonical, so this is not yet very useful.
Now assume that we are in characteristic p > 0, and let ρ : E → E be a degree p morphism as in Section 9 such that ρ * : Consider (ρ, 1) : E ×A 1 → E ×A 1 .The pull-back of S t is a trivial P 1 -bundle for every t ∈ A 1 , so (ρ, 1) * S is a trivial P 1 -bundle over E × A 1 .Pushing forward, we get that |pD| is a basepoint-free relative pencil on S.
We can thus replace 2D with a linear combination , where the G i ∈ |pD| are general and m is large enough.We then have a pair (S, ∆) (we can even make the fibers terminal) such that K S + ∆ ∼ Q 0. Now we have a locally stable morphism where K S + ∆ + D is semiample, and all (large enough) plurigenera jump at t = 0. (If we want terminal fibers, we can also replace D with 1 mp G 1 + • • • + 1 mp G m as above.)Working on the surface S 0 , the Iitaka fibration of K S 0 + ∆ 0 + D 0 is the coordinate projection τ 0 : S 0 E × P 1 → P 1 .However, the restriction of the (relative) Iitaka fibration of S to S 0 is given by a pencil in |pD 0 | whose members are geometrically connected.The only such pencil is the composite of τ 0 with the Frobenius morphism P 1 → P 1 .This example has Kodaira dimension 1, but we can use it to get a general type example in one dimension higher, using the following method.

11.
Let Z be a projective variety whose pluricanonical maps behave in unexpected ways.Then [McK06] constructs a general type pair (of dimension one higher) whose pluricanonical maps also behave unexpectedly.
Choose a sufficiently general embedding Z → P N , and let C ⊂ P N +1 be the cone over Z with vertex v ∈ C.
Blow up v to get Y → C with exceptional divisor E Z.By adjunction, ω Y (E)| E ω E .So, assuming that the maps are surjective, the behavior of the pluricanonical maps of Z should be 'visible' from the pluricanonical maps of the pair (Y , E).
The problem is that Y is a P 1 -bundle over Z, so in fact Second, fix r > 0 not divisible by the characteristic, fix d 1, and let s ∈ O C (drH C ) be a general section.As in [KM98, Section 2.4], these data determine degree r, ramified, cyclic covers We check that the maps are surjective.
Next we go through the details of this for log canonical pairs (Z, ∆ Z ) such that K Z + ∆ Z ∼ Q 0. For these, the cone is also log canonical, and the vertex is a log canonical center.
is semiample and big.We get its canonical model φ Y : Y → Y c .Here φ Y is an isomorphism on Y \ Z 0 , and it restricts to the morphism given by multiples of D Z on Z 0 Z.
with these properties.
If L is very ample, then we can let Z ∞ be the sum of two general members of |Z ∞ |; otherwise, we may need Z ∞ to be a Q-divisor.This may be preferable anyhow since this way we can arrange that (Y , ∆ Y + Z 0 + Z ∞ + D Y ) is terminal away from Z 0 .Note, however, that Z 0 must stay with coefficient 1, so dlt is the best that we can have for (Y , ∆ Y + Z 0 + Z ∞ + D Y ).This is in accordance with [ABL22, Corollary 1.3].

Computing the plurigenera. Let (Y , ∆
for all sufficiently divisible m, we have (13.1) Since D Z is semiample, by Fujita's vanishing theorem [Fuj83] we can choose L sufficiently ample such that for r ≥ 1 and for all sufficiently divisible m.These terms are thus deformation invariant.(In our case, this actually works for any L since Kodaira vanishing holds on the S t ; cf.[Muk13, Theorem 3].) The key term is the r = 0 summand H 0 Z, O Z (mD Z ) of (13.1).Thus we see that if we vary the pair (Z, ∆ Z + D Z ), then any jump in the plurigenera of (Z, ∆ Z + D Z ) leads to the same jump in the plurigenera of In characteristic 0 one usually proves claims (4.c)-(4.d)using inversion of adjunction as in [Kol13, Theorem 4.9].This is not known in positive characteristic; see however [HX15] for related results for 3-folds.The following is a combination of [KSC04, Exercise 6.18] and [Kol13, Proposition 2.7].
Lemma 15.Let X be a regular scheme and i∈I D i a simple normal crossing divisor.For J ⊂ I, set D J := ∩ i∈J D i .Let ∆ be an effective R-divisor.Assume that (15.a) for every J, none of the irreducible components of D J is contained in Supp ∆, and mult x (∆| D J ) ≤ 1 (resp.< 1) for every x ∈ D J .Then (X, D + ∆) is log canonical (resp.dlt ) near D.
Proof.Let E be an exceptional divisor over X.In order to show that a(E, X, D + ∆) ≥ −1, we may localize at the generic point of center X E ⊂ X.So assume that center X E = {x} is a closed point.If x Supp ∆, then we are done by [Kol13, Proposition 2.7].Otherwise, let π : B x X → X be the blow-up with exceptional divisor D 0 .Using induction and [KM98, Lemma 2.45], it is enough to prove the following two claims: The following also shows that the canonical models of the fibers do not form a flat family.
Lemma 17.The (Y i ) c are not dlt, (Y 0 ) c is CM, but (Y 1 ) c (Y c ) t for t 0 is not CM.
Proof.We have τ Y i : Y i → (Y i ) c , which restricts to the elliptic fibration τ i : S i → P 1 .The localization of (Y i ) c at the generic point of its singular set is thus a 2-dimensional, elliptic singularity.In particular, the (Y i ) c are not dlt.
For the CM claims, we aim to use Lemma 18.As in Section 13, the cohomologies of line bundles on the Y i are easy to compute.In particular, we obtain that R 1 (τ Y i ) * O Y i = R 1 (τ i ) * O S i .
With D 1 as in Section 9, τ 1 is given by the pencil |pD 1 , C|.The general member is a smooth, elliptic curve, so R 1 (τ 1 ) * O S 1 is a coherent sheaf of generic rank 1.If it is torsion-free, then it is a line bundle.In particular, H 1 (pD 1 , O pD 1 ) = 1.Next we compute that H 1 (pD 1 , O pD 1 ) = 2, giving a contradiction, so R 1 (τ 1 ) * O S 1 is not torsion-free.
Consider the exact sequence Pushing forward to E and using Claim 9.2, we get Lemma 18.Let g : Y → X be a birational morphism of normal, projective 3-folds.Let L be an ample line bundle on X. Assume that H i (Y , g * L −r ) = 0 for i < 3 and r 1.Then X is CM if and only if R 1 g * O Y does not have a 0-dimensional associated point.

Note that
Proof.The Leray spectral sequence computing H i (Y , g * L −r ) contains Thus H 1 (X, L −r ) = 0, and H 2 (X, L −r ) = 0 for r 1 if and only if R 1 g * O Y has no 0-dimensional associated points.The rest follows from [KM98, Corollary 5.72].
is the tangent map of g * : Pic(E) → Pic(E ) at the origin.(This holds for Abelian varieties; see [Mum70, Section 15].)Thus τ There are two ways to fix this.Let H C be the hyperplane class on C and H Y its pull-back to Y .First, choose d 1, and let D C ∼ dH C be a general divisor not passing through v.It gives D Y ⊂ Y , with D Y disjoint from E, and the pair (Y , D Y + E) is of general type.We work out below that the maps