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(2) − Exc(f) is an ample divisor, Y (3) KY + S is semiample and big, and (4) for every m ≥ 0, the restriction map 0 Y 0 Y H (Y, OY (m(KY + S ))) → H (S , OY (mKSY )), is not surjective. In the last section we give examples of terminal Fano varieties with non-vanishing interme- diate cohomology group by taking the projective cone over Totaro’s example. Theorem 1.3. Let k be an algebraically closed field of characteristic p ≥ 3. Then there exists a Fano variety Z with terminal singularities of dimension 2p +2 over k such that 2 H (Z, OZ ) 6=0. Acknowledgements. I would like to express my gratitude to my advisor Paolo Cascini for his constant support and to Mirko Mauri, Hiromu Tanaka, Burt Totaro and Jakub Witaszek for reading an earlier version of this note and for their insightful comments. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London and Imperial College, London.

2. Preliminaries 2.1. Notation. Throughout this note, k denotes an algebraically closed field. By variety we mean an integral scheme which is separated and of finite type over k. If X is a normal variety, we denote by KX the canonical divisor class. We say that (X, ∆) is a log pair if X is a normal variety, ∆ is an effective Q-divisor and KX + ∆ is Q-Cartier. We say it is log smooth if X is smooth and Supp(∆) is a snc divisor. We refer to [KM98] and [Kol13] for the definition of the singularities appearing in the MMP (e.g. terminal, klt, plt, dlt). We say f : Y → X is a log resolution if f is a birational morphism, Y is regular, Exc(f) has pure codimensione one and the pair (Y, Supp(f −1∆ + Exc(f))) is log smooth. We denote by ρ(X) := ρ(X/ Spec k) the Picard number of X.

2.2. Fano varieties violating Kodaira vanishing in positive characteristic. While coun- terexamples to Kodaira vanishing for general projective varieties in positive characteristic are quite abundant in literature (see for example [Ray78]), until the recent work of Totaro (see [Tot17]) only few examples of smooth Fano varieties violating the Kodaira vanishing theorem were known (see [LR97] and [Kov17]). We recall the properties of Totaro’s example we need to prove Theorem 1.1: Theorem 2.1 ([Tot17, Theorem 2.1]). Let k be an algebraically closed field of characteristic p ≥ 3. Then there exists a smooth Fano variety X over k of dimension 2p+1 with a very ample Cartier divisor A such that (1) ρ(X)=2, (2) −KX =2A and 1 (3) H (X, OX (A)) 6=0. This theorem has already been a prolific source of examples of pathologies in positive char- acteristic: by taking the cone over X, Totaro shows that for every p ≥ 3 there exists a terminal not Cohen-Macaulay singularity in dimension 2p +2 ([Tot17, Corollary 2.2]) and in [AZ17] the authors construct non-liftable Calabi-Yau varieties by considering anticanonical sections of X and double covers along a general member of |− 2KX |. 2.3. Affine cones. For the theory of cones over algebraic varieties and the study of their singularities we refer to [Kol13, Chapter 3]. Here we recall only what we need to prove our main result. Let (X, ∆) be a log pair over k and let L be an ample line bundle on X. We denote by C (X,L) := Spec H0(X,L⊗m) the cone over X induced by L and by ∆ a X Pm≥0 Ca(X,L) NON-NORMAL PLT CENTRES IN CHARACTERISTIC p ≥ 3 3

1 the induced Q-divisor on Ca(X,L). Over X we consider the A −bundle π : BCa(X,L) := ⊗m SpecX Pm≥0 L → X and we have the following diagram: π / BCa(X,L) X

f  Ca(X,L) , where f is the birational morphism contracting the section X− of π with anti-ample normal − ∨ bundle NX /BCa(X,L) ≃ L . The following describes the divisor class group of the cone and the condition under which its canonical class is Q-Cartier: Proposition 2.2 ([Kol13, Proposition 3.14]). With the same notation as above, we have (1) Cl(Ca(X,L)) = Cl(X)/hLi, (2) Pic(Ca(X,L))=0, and

(3) m(KCa(X,L)+∆Ca(X,L)) is Cartier if and only if there exists r ∈ Q such that OX (m(KX + ∆)) ≃ Lrm. From the point of view of the singularities of the MMP we have the following Proposition 2.3 ([Kol13, Lemma 3.1]). With the same notation as above, let us assume that KX + ∆ ∼Q rL. Then we have ∗ − ∗ (1) KBCa(X,L) + π ∆+(1+ r)X = f (KCa(X,L) + ∆Ca(X,L)); and the pair (Ca(X,L), ∆Ca(X,L)) is terminal (resp. dlt) if and only if the pair (X, ∆) is terminal (resp. dlt) and r< −1 (resp. r< 0). 2.4. Projective cones. Let X be a normal variety over k and let L be an ample line bundle on X. We define the projective cone of X with respect to L as in [Kol13, Section 3.8]: m 0 r m−r Cp(X,L) = Projk X (X H (X,L )xn+1 ). m≥0 r

It contains as an open dense subset Ca(X,L) and it admits a partial resolution

π : PX (OX ⊕ L) → Cp(X,L).

We show how to compute the cohomology of the structure sheaf of Cp(X,L) in terms of the cohomology groups of L on X: Proposition 2.4. For i ≥ 2, i i−1 m H (Cp(X,L), OCp(X,L)) ≃ X H (X,L ). m>0

Proof. We denote by v the vertex of Ca(X,L) ⊂ Cp(X,L) and we consider the natural inclusion m m V := Ca(X,L) \ v = SpecX X L ⊂ SpecX X L = Cp(X,L) \ v =: U. m∈Z m≤0 Considering the long exact sequences in local cohomology (see [Har77, Chapter III, Ex 2.3]) we have the following natural commutative diagram

/ i−1 δi / i / i / ... H (U, OU ) Hv(Cp(X,L), OCp(X,L)) H (Cp(X,L), OCp(X,L)) ...

∗ ∗ i ≃ i    / i−1 ηi / i / i / ... H (V, OV ) Hv(Ca(X,L), OCa(X,L)) H (Ca(X,L), OCa(X,L)) ..., where by i we mean the natural inclusion maps. It is easy to see that the diagram

∗ i i / i H (U, OU ) H (V, OV )

≃ ≃   i−1 m / i−1 m Pi≤0 H (X,L ) Pm∈Z H (X,L ) 4 FABIO BERNASCONI commutes, where the bottom arrow is the natural inclusion, thus showing that i∗ is injective. i−1 Since for i ≥ 2 the maps ηi are isomorphisms (due to the vanishing H (Ca(X,L), OCa(X,L))= 0), we conclude that δi are injective and thus we have the following commutative diagram of short exact sequences:

// i−1 δi / i / i / 0 H (U, OU ) Hv(Cp(X,L), OCp(X,L)) H (Cp(X,L), OCp(X,L)) 0

≃ ≃ ≃    0 / Hi−1(X,Lm) / Hi−1(X,Lm) /Hi(C (X,L), O ) /0, Pm≤0 Pm∈Z p Cp(X,L) which concludes the proof. 

3. Non-normal plt centres To prove the main result we consider a cone over the Fano variety X constructed by Totaro and we show that the prime divisor induced on the cone by a smooth section E ∈ |A| is not normal.

Proof of Theorem 1.1. Let us fix an algebraically closed field k of characteristic p ≥ 3. Let us consider the smooth Fano variety X with the ample divisor A of Theorem 2.1. Since A is very ample, by Bertini theorem there exists a smooth divisor E ∈ |A|. We define the pair

(Z,S) := (Ca(X, OX (A)), ECa(X,OX (A))). Since (X, E) is log smooth and KX + E ∼ −A we conclude by Proposition 2.2 that KZ + S is Cartier and by Proposition 2.3 that the pair (Z,S) is dlt. Since S is the only irreducible component in the boundary, the pair (Z,S) is actually Y −1 plt. We denote by (Y,S ) the pair given by (BCa(X, OX (A)),f∗ S). We check that S is not normal. Let us note that 0 0 S = Speck X Im(H (X, OX (mA)) → H (E, OE(mA)). m≥0 Let us consider the short exact sequence

0 → OX (−E) → OX → OE → 0, and tensor it with nA, where n is a positive integer. Taking the long exact sequence in coho- mology we have 0 0 1 1 H (X, OX (nA)) → H (E, OE (nA)) → H (X, OX ((n − 1)A)) → H (X, OX (nA)). Since A is ample and H1(X, O(A)) 6=0, we can consider, by Serre vanishing, the largest n ≥ 2 1 1 such that H (X, OX ((n − 1)A)) 6=0 and H (X, OX (nA))=0. Thus the morphism 0 0 H (X, OX (nA)) → H (E, O(nA)) is not surjective and therefore the natural morphism ν ν : S := Ca(E,L|E) → S, is not an isomorphism. The morphism ν is finite and it is birational since for m>n we 0 0 0 have H (E, OE(mA)) = Im(H (X, OX (mA)) → H (E, OE (mA)). Thus we conclude that the variety S is not normal and that ν is the normalisation morphism, since Sν is normal.  Remark 1. We note that, since ρ(X)=2, the affine variety Z is not Q-factorial by Proposition 2.2. Remark 2. We point out that S is regular in codimension one and thus by Serre’s criterion we deduce that S does not satisfy the S2 condition. In general, plt centres satisfy the R1 condition also in positive characteristic as explained in [GNT15, Lemma 2.5]. Remark 3. The normalisation morphism ν : Sν → S is a universal homeomorphism since the morphism SY → S has connected fibers. In [GNT15, Theorem 3.15], the authors prove, assuming the existence of pl-flips, that this is always the case for plt centres on threefolds in any characteristic. NON-NORMAL PLT CENTRES IN CHARACTERISTIC p ≥ 3 5

Proof of Corollary 1.2. We verify that properties (1)−(3) hold. The first property is immediate by construction of (Y,SY ) as an A1-bundle over a log smooth pair. To show property (2) we note that the divisor −X− is f-ample over the affine variety Z, thus it is ample. We have, by formula (1) of Proposition 2.3, that Y ∗ KY + S = f (KZ + S), and since KZ + S is ample on Z we conclude property (3) holds. We now show property (4). Since S is not normal we have that the morphism

(2) f∗OY → f∗OSY is not surjective. Indeed, since SY → Sν is a surjective birational projective morphism between normal varieties, we have by Zariski’s main theorem the following commutative diagram: / / OZ OS ❏ ❏❏ ❏❏ ❏❏ ❏❏% ≃ ν∗OSν t ≃ tt tt tt   tz t / f∗OY f∗OSY .

Since OS → ν∗OSν is not surjective, we conclude that the bottom arrow is not surjective. By Proposition 2.2 we have that KZ +S ∼ 0 and thus, using the projection formula, we have Y ∗ f∗OY (m(KY + S )) = f∗(f OZ (m(KZ + S))) = f∗OY , and in the same way

f∗OSY (mKSY )= f∗OSY . Since Z is affine, we have therefore that the morphism in (2) is not surjective if and only if 0 Y 0 Y H (Y, OY (m(KY + S ))) → H (S , OSY (mKSY )), is not surjective for any integer m> 0, thus concluding the proof. 

2 4. Terminal Fano varieties with H (OZ ) 6=0 In this section we prove Theorem 1.3. Since we need to understand the positivity properties of the canonical divisor of a projective bundle we recall the Euler sequence. Let E be a vector bundle of rank r +1 on a variety X and let π : PX (E) → X be the associated projective bundle of hyperplanes, then we have the following short exact sequence: 1 ∗ 0 → ΩP(E)/X → OP(E)(−1) ⊗ π E → OP(E) → 0, ∗ which shows KP(E) = OP(E)(−r − 1) ⊗ π (KX ⊗ det E). Proof of Theorem 1.3. Let us fix an algebraically closed field k of characteristic p ≥ 3 and let us consider the Fano variety X with the ample divisor A of Theorem 2.1. We define the 1 2 projective variety Z := Cp(X, OX (A)). Since H (X, OX (A)) 6=0, we conclude H (Z, OZ ) 6=0 by Proposition 2.4. The variety Z has terminal singularities since the only singular point is the vertex of the cone Ca(X, OX (A)). We are only left to prove that Z is a Fano variety. For this it is sufficient to check that the projective bundle π : Y := PX (OX ⊕ OX (−A)) → X is Fano. On Y there is a unique negative − − + section X such that OY (X )|X− ≃ OX (−A). There exists also a positive section X such + + that OY (X )|X+ ≃ OX (A) (which implies that X is a big and nef divisor on Y ) and such that X+ ∼ X− + π∗A. We note that − OY (1) = OY (X ), and by the relative Euler sequence we have ∗ KY = OY (−2) ⊗ π (KX − A). 6 FABIO BERNASCONI

Thus we have that the anticanonical class − ∗ + ∗ (3) − KY =2X + π (3A)=2X + π A, is a big and nef divisor. To conclude that −KY is ample, we show that the null locus Null(−KY ) (see [Laz04, Definition 10.3.4]) is empty. By equation (3), the null locus must be contained in − −  X , but since −KY |X = A is ample we conclude it must be empty.

5. Open questions A question that naturally arises from Theorem 1.1 and from the result of [HX15] for threefolds in characteristic p> 5 is the following:

Question 1. Fixed a positive integer n, does there exists a number p0(n) such that n- dimensional plt centres in characteristic p ≥ p0(n) are normal? A guiding philosophy in the study of MMP singularities in large characteristic is that their good behaviour should correspond to Kodaira-type vanishing results for (log) Fano varieties (as an instance of this principle see the main result of [HW17]). If one is interested in developing the MMP for threefolds in characteristic p ≤ 5, it would be important to know an optimal bound for p0(3). Since we know by [CT16, Theorem 1.1] that there exists a threefold plt pair with non-normal centre in characteristic two we ask the following Question 2. Are plt threefold centres normal in characteristic three and five? In [Ber17, Theorem 2] we constructed an example of a klt not Cohen-Macaulay threefold singularity in characteristic three. This suggests that more pathologies for threefold singularities could appear in characteristic three.

References

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Department of Mathematics, Imperial College, London, 180 Queen’s Gate, London SW7 2AZ, UK E-mail address: [email protected]