Supersingular quartic surfaces

Journal of Pure and Applied Algebra 223 (2019) 4701–4707

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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa

Supersingular quartic surfaces Junmyeong Jang a,b,∗ a

Department of Mathematics, University of Ulsan, Daehakro 93, Namgu Ulsan 44610, Republic of Korea School of Mathematics, Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul 02455, Republic of Korea b

a r t i c l e

i n f o

Article history: Received 19 June 2018 Received in revised form 28 November 2018 Available online 20 February 2019 Communicated by S. Kovacs

a b s t r a c t In this paper, we prove that, over an algebraically closed field of odd characteristic p (p = 5), every supersingular K3 surface is isomorphic to a quartic surface in the three dimensional projective space. © 2019 Elsevier B.V. All rights reserved.

MSC: 14J28 Keywords: Supersingular K3 surface Quartic polarization

1. Introduction Throughout this article, we assume k is an algebraically closed field of characteristic p = 2 unless stated otherwise. A smooth quartic surface in P3k is a K3 surface. Let X be a smooth quartic surface over k. The line bundle of linear sections O(1) is an ample line bundle on X and the self intersection O(1) · O(1) = 4. Conversely, if L is an ample line bundle on a K3 surface X of self intersection 4, the linear system |L| defines a morphism ϕL : X → P3 . If ϕL is an embedding, Im ϕL is a smooth quartic surface. Otherwise, Im ϕL is a smooth quadric surface and ϕL is a double cover. ([2], p. 89) Assume ϕL is a double covering and let Q = Im ϕL . Then Q is isomorphic to P1 ×P1 andN S(Q)  U . Here U is an even unimodular hyperbolic lattice of rank 0 1 2 with a matrix presentation . In this case, L ∈ ϕ∗L (N S(Q))  U (2) and ϕ∗L (N S(Q)) is a primitive 1 0 sublattice of N S(X).

* Correspondence to: Department of Mathematics University of Ulsan, Daehakro 93, Namgu Ulsan 44610, Republic of Korea. E-mail address: [email protected]. https://doi.org/10.1016/j.jpaa.2019.02.012 0022-4049/© 2019 Elsevier B.V. All rights reserved.

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Proposition 1.1. Assume X is a K3 surface defined over k and L is an ample line bundle on X with self intersection L · L = 4. In this assumption, L is very ample if and only if there does not exist a primitive sublattice M ⊂ N S(X) which is isomorphic to U (2) and L ∈ M . Proof. In the above, we have observed that, if Im ϕL is a smooth quadric surface, L ∈ ϕ∗L (N S(Im ϕL ))  U (2). Now assume M is a primitive sublattice of N S(X) and M is isomorphic to U (2). Let e, f be a basis of M such that e · e = f · f = 0, e · f = 2 and L = e + f . Then the linear systems of e and f do not have a fixed component and define genus 1 fibrations ϕe : X → P1 and ϕf : X → P1 . Moreover H 0 (X, L) = H 0 (X, e) ⊗ H 0 (X, f ). Therefore ϕL = ϕe × ϕf and Im ϕL is isomorphic to P1 × P1 . 2 Whether a degree 4 ample line bundle on a K3 surface gives a quartic surface structure is determined by the position of the class of the line bundle in the Neron-Severi lattice. For a K3 surface X over k, let Δ = {v ∈ N S(X)|v · v = −2} ⊂ N S(X). For each v ∈ Δ, let sv (∈ O(N S(X))) be the reflection along the line of v, sv (w) = w + (v · w)v. The Weyl group WX of N S(X) is the subgroup of O(N S(X)) generated by all sv , (v ∈ Δ) and −id. A vector w ∈ N S(X) is the class of an ample line bundle if and only if w · w > 0 and w · v > 0 for all effective v ∈ Δ. Moreover if w ∈ N S(X) satisfies that w · w > 0 and w · v = 0 for any v ∈ Δ, there exists a unique ψ ∈ WX such that ψ(w) is the class of an ample line bundle. ([13], 1.10)  X is a smooth Assume k is of characteristic p > 2. The formal Brauer group of a K3 surface X over k, Br  1-dimensional formal group over k. The height of the formal Brauer group BrX is an integer between 1 and 10 or the infinite.  X is finite h, the rank of the Neron-Severi lattice N S(X) is at most 22 − 2h. ([6], If the height of Br Proposition 5.12) A K3 surface of height 1 is an ordinary K3 surface. A K3 surface X is ordinary if and only if the absolute Frobenius morphism on the second cohomology of the structure sheaf, ∗ : H 2 (X, OX ) → H 2 (X, OX ) FX

is non-zero.  X is the infinite. A K3 surface A K3 surface X over k is a supersingular K3 surface if the height of Br X is supersingular if and only if the rank of N S(X) is 22. ([4], [9]) Also, if p > 3, a K3 surface X is supersingular if and only if X is unirational. ([8]). Every supersingular K3 surface X has a nef line bundle of self intersection 2 which induces the following morphism X → Y → P2 , here Y → P2 is a double covering ramified on a sextic curve and X → Y is a desingularization of rational double points. ([14]) Assume X is a supersingular K3 surface. The discriminant group of N S(X), l(N S(X)) = N S(X)∗ /N S(X) is isomorphic to (Z/p)2σ for an integer σ between 1 and 10. We call σ is the Artin-invariant of X. The discriminant of the quadratic form imposed on l(N S(X)) is (−1)σ Δ. Here Δ is a non quadratic residue modulo p. An even integral lattice of signature (1, 21) with the discriminant group isomorphic to (Z/p)2σ of discriminant (−1)σ Δ is unique up to isomorphism. ([10], Proposition 1.14.1) Therefore, the Neron-Severi lattice of a supersingular K3 surface is determined by the base characteristic and the Artin-invariant. When X and X  are supersingular K3 surfaces over k of same Artin-invariant, then N S(X) is isomorphic to N S(X  ). For a supersingular K3 surface of Artin invariant σ, N S(X) ⊗ Zp = E0 ⊕ E1 (p),

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where E0 and E1 are unimodular Zp -lattices of rank 22-2σ and 2σ respectively. ([12], Proposition 3.13) All the supersingular K3 surfaces of Artin invariant σ form a σ − 1 dimensional family. A supersingular K3 surface of Artin invariant 1 is unique up to isomorphism. In this paper, we prove that when the base field k is of odd characteristic p and p = 5, every supersingular K3 surface over k is isomorphic to a smooth quartic surface in P3 . 2. Main result Assume k is an algebraically closed field of odd characteristic p. Let W be the ring of Witt vectors of k and K be the fraction field of W . Lemma 2.1. Assume k is an algebraically closed field of the same characteristic with k. Suppose X/k and X  /k are supersingular K3 surfaces of same Artin-invariant σ. 1. If X has a polarization of level 2n, then so does X  . 2. If X is isomorphic to a quartic surface, then so is X  . Proof. We fix an isomorphism f : N S(X) → N S(X  ). Let L be an ample line bundle on X of self intersection number 2n. Then f (L) · f (L) = 2n and f (L) · e = 0 for any e ∈ N S(X  ) such that e · e = −2. Hence there exists ψ ∈ WX  such that ψ(f (L)) is an ample class of level 2n. Assume L is a very ample line bundle on X of self intersection 4 which induces an embedding ϕL : X → P3 . We assume ψ(f (L)) is an ample class in N S(X  ) for a ψ ∈ WX  . If a primitive sublattice M of N S(X  ), which is isomorphic to U (2), contains ψ(f (L)), L is contained in ψ −1 (f −1 (M ))  U (2). This is contradiction to that ϕL is an embedding by Proposition 1.1. Hence ψ(f (L)) induces an embedding of X  into P3 again by Proposition 1.1. 2 Lemma 2.2. Let X be a supersingular K3 surface of Artin invariant 1 over k. 1. If X has a polarization of level 2n, then so does every supersingular K3 surface over k. 2. If X is isomorphic to a quartic surface in P3 , then so is every supersingular K3 surface over k. Proof. Let S = Spf k[[t1 , · · · , t20 ]] be the deformation space of X over artin local k-algebras. Assume L is an ample line bundle of self intersection number 2n on X. Let Σ(L) ⊂ S be the locus of deformations to which the line bundle L extends. The locus Σ(L) is defined by one equation in S and is of 19 dimensional. ([12], Proposition 2.2) By the existence theorem and some modification, we may assume the locus Σ(L) and the family X → Σ(L) are scheme of finite type over k. The extension of L gives an ample line bundle of self intersection 2n on each fiber of X /Σ(L). Let x ∈ Σ(L) be the central closed point corresponding to (X, L). Let Σ(L) = M1 ⊃ M2 ⊃ · · · ⊃ M10 ⊃ Σ10 ⊃ Σ9 ⊃ · · · ⊃ Σ1 be the height-Artin invariant stratification of Σ(L). Here Mi is the reduced closure of the locus of height i fibers and Σj is the reduced closure of the locus of supersingular fibers of Artin invariant j. In this stratification, each step is defined by one equation, so each step is of codimension at most 1. ([1], p. 563) Since X is of Artin invariant 1, x ∈ Σ1 . But a supersingular K3 surface is unique up to isomorphism, so Σ1 is of dimension 0. Because M1 is 19 dimensional and there are 20 steps in the stratification, each step is of codimension 1. In particular, each strata Σj is purely j − 1 dimensional. We choose a point η in Σj − Σj−1 (2 ≤ j ≤ 10) such that x is in the closure of η. Let Xη¯ be a geometric fiber over η. Xη¯ is a supersingular K3 surface of Artin invariant of j over k(η) and has an ample line bundle of self intersection

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2n, L. Therefore, by Lemma 2.1, every supersingular K3 surface of Artin invariant j over k has a polarization of level 2n. Now assume L · L = 4 and ϕL is an embedding of X into P3 . Let 2 c : N S(X) → Hcris (X/W )

be the cycle map and 2 2 π : Hcris (X/W ) → Hdr (X/k) 2 be the mod p reduction map. Since the rank 1 sublattice of Hcris (X/W ) generated by c(L) is unimodular, there is a decomposition 2 Hcris (X/W ) =< c(L) > ⊕ < c(L) >⊥ . 2 2 Let F : Hcris (X/W ) → Hcris (X/W ) be the canonical Frobenius semi-linear morphism on the crystalline cohomology induced by the absolute Frobenius morphism of X. Because 2 Hcris (X/W )F =p = c(N S(X)) ⊗ Zp , ([12], Corollary 1.6) 2 2 2 we have F (c(L)) = pc(L). But F 2 Hdr (X/k) = π(F −1 (p2 Hcris (X/W )), so F 2 Hdr (X/k) ⊂ π(< c(L >⊥ ). 2 Therefore, π(c(L)) ∈ / F 2 Hdr (X/k) and Σ(L) is smooth at x. ([12], Proposition 2.2) We may assume Σ(L) is smooth scheme over k. Then for a point η ∈ Σj − Σj−1 chosen above, there exists a reduction map N S(Xη¯) → N S(X) and the following diagram commutes.

He´2t (Xη¯, Ql ) 6

∪

∼- 2 He´t (X, Ql ) 6

∪

N S(Xη¯) ⊗ Ql - N S(X) ⊗ Ql . Here two vertical cycle maps are injective. Since N S(Xη¯) is torsion free, the reduction map N S(Xη¯) → N S(X) is injective. (cf. [16], Proposition 6.3) Therefore L ∈ N S(Xη¯) ⊂ N S(X). If M  U (2) is a primitive sublattice of N S(Xη¯) and L ∈ M , then M is also a primitive sublattice of N S(X) since N S(X)/N S(Xη¯) is a p-group. It is a contradiction to that ϕL is an embedding. It follows that Xη¯ is isomorphic to a quartic surface. By Lemma 2.1, every supersingular K3 surface of Artin invariant j is a quartic surface. 2 Remark 2.3. It is known that the cycle class of Σ1 is a scalar multiplication of a power of the class of the Hodge bundle H 0 (X , Ω2X /Σ(L) ). ([5]) But it is not guaranteed that the locus Σ1 is not empty. Theorem 2.4. If p = 5, every supersingular K3 surface over k is isomorphic to a quartic surface in P3 . Proof. By Lemma 2.1, it is enough to show that a supersingular K3 surface of Artin invariant 1 is isomorphic to a quartic surface. We assume F is an arbitrary algebraically closed field of characteristic = 2. Let fλ = x4 + y 4 + z 4 + w4 + λxyzw, λ ∈ F and Xλ be the quartic surface defined by fλ in P3F = Proj F [x, y, z, w]. The family of quartic surfaces {Xλ } is called Dwork pencil. It is easy to check that Xλ is non-singular if and only if λ4 = 256. When F = C and λ is transcendental over Q, the Neron-Severi lattice N S(Xλ ) is of rank 19 and the discriminant group of N S(Xλ ) is isomorphic to (Z/8)2 × (Z/4). ([3], 4.1.) Let us denote a lattice

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isomorphic to such N S(Xλ ) by C. It is also known that for any λ ∈ C, there is an embedding C → N S(Xλ ). ([3], 4.1.) Because there exists a unique supersingular K3 surface of Artin invariant 1 up to isomorphism over arbitrary k, we may assume that the cardinality of k is equal to or less than the cardinality of R. Let K be ¯  C. Let λ ∈ k, λ4 = 256 the fraction field of the ring of Witt vectors W . We fix an arbitrary isomorphism K ¯ As we observed and Λ be a lifting of λ in W . Then Λ4 = 256 and XΛ is a smooth quartic surface over K. above, there is an embedding of C into N S(XΛ ) and N S(XΛ ) is a sublattice of N S(Xλ ). Therefore, the rank of N S(Xλ ) is at least 19 and C ⊗ Zp is a unimodular sublattice of rank 19 of N S(X) ⊗ Zp . Assume Xλ is of finite height h. Since 19 = rank C ≤ rank N S(X) ≤ 22 − 2h, h = 1 and X is ordinary. Assume Xλ is supersingular of Artin invariant σ. The unimodular part E0 of N S(X) ⊗ Zp is of rank 22 − 2σ. Since C ⊗ Zp is unimodular, rank C ≤ rank E0 = 22 − 2σ and σ = 1. Therefore Xλ is ordinary or supersingular of Artin invariant 1. Recall that Xλ is ordinary if and only if ∗ : H 2 (Xλ , OXλ ) → H 2 (Xλ , OXλ ) FX λ ∗ is non-zero. Here FX : OXλ → OXλ is the absolute Frobenius morphism. Let us consider the following λ diagram.

0

- O(−4)

FP∗3 0

- OX λ

FP∗3

? fp ? - O(−4p) λ - OP 3

fλp−1 0

fλ OP3

? - O(−4)

- 0

FP∗3 ? - OX λ,p

- 0

? ? - OX λ

- 0

fλ - ? OP3

Here Xλ,p is the non-reduced surface defined by fλp . The composition of two right vertical arrows is the Frobenius morphism of OXλ . From the above diagram, we obtain the following. H 2 (OXλ )

∼- 3 H (O(−4)) FP∗3

? ? ∼ H (OXλ,p ) - H 3 (O(−4p)) 2

1 xyzw ? 1 (xyzw)p

fλp−1 ? ? ∼ H 2 (OXλ ) - H 3 (O(−4))

?

fλp−1 (xyzw)p

Because the non-vanishing classes in H 3 (P3 , O(−4)) are generated by

∗ the Frobenius morphism FX λ

on H 2 (Xλ , OXλ ) is zero if and only if the coefficient of (xyzw)p−1 in

0. This coefficient is equal to

1 xyzw , p−1 fλ is

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φp (λ) =

 i=0

(p − 1)! λp−1−4i . (p − 1 − 4i)!(i!)4

If p ≡ 3 mod 4, λ = 0 is a solution of φp (λ) = 0 and the Fermat quartic X0 is supersingular of Artin invariant 1. (cf. [15], p. 587) Assume p ≡ 1 mod 4. Let t = λ4 and put ψp (t) = φp (λ). If t = 256 is p−1 = (p − 1)(p − 2)(p − 3)(p − 4), the only solution of ψp (t) = 0, then ψp (t) = (t − 256) 4 and − 256(p−1) 4 equivalently, 64 ≡ 24 mod p. It happens only when p = 5. Therefore, if p = 5, for some λ ∈ k, Xλ is a smooth supersingular quartic surface of Artin invariant 1. This completes the proof. 2 ¯ N S(Xλ ) = C Remark 2.5. As we recall above, Bini and Garbagnati prove that when k = C and λ ∈ C − Q, ∗ 2 for a lattice C of rank 19 and with the discriminant group C /C = (Z/8) × (Z/4). When k is of odd ¯ p ⊂ k, by the Tate conjecture ([11]), the rank of N S(Xλ ) is even. In particular, characteristic and λ ∈ F ¯ p . Then Xλ is an ordinary smooth quartic if Xλ is ordinary, the rank of N S(Xλ ) is 20. Assume λ ∈ k − F surface and the rank of N S(Xλ ) is 19 or 20. The quartic surface Xλ is isomorphic to a geometric generic ¯ p and for any s ∈ F¯p , there is an injective fiber of the Dwork family defined over the polynomial ring F reduction map N S(Xλ ) → N S(Xs ). Assume the rank of N S(Xλ ) is 20 and d = N S(Xλ ). Then for almost ¯ p , N S(Xs ) is of rank 20 with the discriminant dividing d. But there are only finitely many K3 all s ∈ F surfaces up to isomorphism of Picard number 20 with bounded discriminant of the Neron-Severi lattice. ([7], Theorem 3.7.) Since Dwork family is non-isotrivial, it is a contradiction and the rank of N S(Xλ ) is 19. Let us choose Λ ∈ W , a lifting of λ, which is transcendental over Q. Then for a quartic surface ¯ N S(XΛ ) = C and the reduction map N S(XΛ ) → N S(Xλ ) is an embedding of lattices of XΛ over K, same rank. In this case, the cokernel N S(Xλ )/N S(XΛ ) is a finite p-group. But C ⊗ Zp is unimodular, so N S(Xλ ) = N S(XΛ ) = C. Remark 2.6. Over an arbitrary algebraically closed field of characteristic p, there exists a unique supersingular K3 surface of Artin invariant 1 up to isomorphism. In particular, there exists a unique supersingular K3 surface of Artin invariant 1 over an algebraic closure of a finite field Fp . Hence a unique supersingular K3 surface of Artin invariant 1 has a model defined over a finite field. When p ≡ 3 mod 4, the Fermat quartic ¯p surface X0 is a model of a supersingular K3 surface of Artin invariant 1 over the prime field. Let k = F r and X be a supersingular K3 surface of Artin invariant 1 defined over k. Let X (p ) be the base change of X by the rth power of the Frobenius morphism of k, Fkr . X (p

r

)

? k

Fkr X

Fkr - ? k m

Assume X is defined over a finite field Fpm for a positive integer m. In this case, X (p ) = X. Because a supersingular K3 surface of Artin invariant 1 is unique up to isomorphism, for any positive integer r, there r exists an isomorphism g : X (p ) → X over k. Suppose r is an divisor of m. By the Galois descent, X has a r model over Fpr if there exists an isomorphism g : X (p ) → X such that (m−r)∗

g ◦ Fkr∗ (g) ◦ Fk2r∗ (g) ◦ · · · ◦ Fk

(g) : X → X

is equal to the identity. The discriminant group l(N S(X)) = N S(X)∗ /N S(X) is isomorphic to (Z/p)2 and is equipped with the induced quadratic form. The period space of X,

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2 K(X) = Hcris (X/W )/(N S(X) ⊗ W )

is an isotropic line in (N S(X)∗ ⊗ W )/(N S(X) ⊗ W ) = l(N S(X)) ⊗ k. Let FW : W → W be the Frobenius r 2 2 r∗ (X (p ) /W ) = Hcris (X/W ) ⊗W FW W , if morphism on W . Let f = id ⊗ Fk : l ⊗ k → l ⊗ k. Since Hcris r r (p ) r∗ (p ) (pr ) ) by the isomorphism Fk : N S(X) → N S(X ), we have K(X )= we identify N S(X) and N S(X 2 f r (K(X)) in l ⊗ k. In particular, K(X (p ) ) = K(X) and the isomorphism 2

Fk2∗ : N S(X) → N S(X (p ) ) preserves the ample cone and the period space. Hence, by the crystalline Torelli theorem, ([13], Theorem II) 2 there exists an isomorphism over k, α : X (p ) → X such that α∗ |N S(X) = Fk2∗ |N S(X). This isomorphism α satisfies the cocycle condition of the Galois descent and X has a model defined over Fp2 . Assume p ≡ 1 mod 4 and e = λp−1 is a fourth root of unity for some λ ∈ k. Then λ4(p−1) = tp−1 = 1 and t ∈ Fp . The isomorphism (p)

Xλ = Xλp → Xλ , (x, y, z, w) → (x, y, z, ew) satisfies the cocycle condition and Xλ has a model over the prime field Fp . Therefore if the equation ψp (t) = 0 has a solution in Fp which is different from 256, a supersingular K3 surface of Artin invariant 1 has a model over the prime field. Acknowledgement The author appreciates to Professor F. Catanese for suggesting a simple argument of the proof of Proposition 1.1. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [2018R1D1A1B07044995]. References [1] M. Artin, Supersingular K3 surfaces, Ann. Sci. Éc. Norm. Supér. (4) (7) (1974) 543–567. [2] A. Beauville, J.-P. Bourguignon, M. Demazure, Géomeétrie des Surfaces K3: Modules et Périodes, Astérisque, vol. 126, Soc. Math. de France, 1985. [3] G. Bini, A. Garbagnati, Quotients of the Dwork pencil, J. Geom. Phys. 75 (2014) 173–198. [4] F. Charles, The Tate conjecture for K3 surfaces over finite fields, Invent. Math. 194 (2013) 119–145. [5] T. Ekedahl, G. van der Geer, Cycle classes on the moduli of K3 surfaces in positive characteristic, Sel. Math. New Ser. 21 (2015) 245–291. [6] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979) 501–661. [7] J. Jang, Neron-Severi group preserving lifting of K3 surfaces and applications, Math. Res. Lett. 22 (2015) 289–802. [8] C. Liedtke, Supersingular K3 surfaces are unirational, Invent. Math. 200 (2015) 979–1014. [9] K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015) 625–668. [10] V.V. Nikulin, Integral symmetric bilinear forms and some of their applications, Izv. Akad. Nauk SSSR 43 (1979) 111–177. [11] N. Nygaard, The Tate conjecture for ordinary K3 surfaces over finite fields, Invent. Math. 74 (1983) 213–237. [12] A. Ogus, Supersingular K3 crystal, Astérisque 64 (1979) 3–86. [13] A. Ogus, A crystalline torelli theorem for supersingular K3 surfaces, Prog. Math. 36 (1983) 361–394. [14] I. Shimada, Supersingular K3 surfaces in odd characteristic and sextic double planes, Math. Ann. 328 (2004) 451–468. [15] T. Shioda, in: Supersingular K3 Surfaces, Algebraic Geometry, Proc. Summer Meeting, Univ. Copenhagen, in: LNS, vol. 732, 1979, pp. 564–591. [16] R. van Luijk, An elliptic K3 surfaces associated to Heron triangles, J. Number Theory 123 (2007) 92–119.