Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

We prove that any Fourier–Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the set of Fourier–Mukai partners of a canonical cover of a hyperelliptic or Enriques surface over an algebraically closed field of characteristic greater than three is trivial. These results extend earlier results of Bridgeland–Maciocia and Sosna to positive characteristic.


Introduction
The main motivation of this paper is the recent series of results in the study of equivalences of derived categories of sheaves of smooth projective varieties over fields other than the field of complex numbers. For instance, over finite fields, the first named author proves that the Hasse-Weil zeta function of an abelian variety, as well as of smooth varieties of dimension at most three, is unaltered under equivalences of derived categories [9,11]. Moreover, Ward in his thesis [35] produces examples of genus one curves over Q admitting an arbitrary number of distinct Fourier-Mukai partners, revealing in this way consistent differences with the case of elliptic curves over C. Finally, Lieblich and Olsson in [17] extend to positive characteristic seminal works of Mukai and Orlov concerning derived equivalences of K 3 surfaces. In particular, they prove that any Fourier-Mukai partner of a K 3 surface X over an algebraically closed field of characteristic p = 2 is a moduli space of Gieseker-stable sheaves on X , and in addition X admits only a finite number of Fourier-Mukai partners. While Orlov's proof relies on Hodge theory, Lieblich-Olsson's proof relies on deformation theory of perfect complexes and on the theory of liftings to the Witt ring.
Inspired by the results of [17], in this paper we focus on special classes of abelian and K 3 surfaces that arise as canonical covers of hyperelliptic and Enriques surfaces. Our first main result is an extension of a work of Sosna [34,Theorem 1.1] to positive characteristic. Theorem 1.1 Let S be a hyperelliptic surface over an algebraically closed field of characteristic p > 3 and let A be its canonical cover. Then any smooth projective surface that is derived equivalent to A is isomorphic to either A or its dual A.
We refer to Theorem 4.4 for a slightly stronger result. One of the main ingredients in the proof of Theorem 1.1 is the characterization of Fourier-Mukai partners of abelian surfaces in positive characteristic as moduli spaces of Gieseker-stable sheaves. In particular we extend the result [6, Theorem 5.1] of Bridgeland-Maciocia to positive characteristic. In the following we denote by D(X ) the bounded derived category of coherent sheaves on a smooth projective variety X .

Theorem 1.2 Let A be an abelian surface over an algebraically closed field k of positive characteristic and let Y be a smooth projective variety over k. Suppose furthermore that there is an equivalence of triangulated categories : D(A) → D(Y ). Then Y is an abelian surface and A is isomorphic to a moduli space of Gieseker-stable sheaves on either Y or its dual Y .
The proof of the previous theorem is based on the notion of filtered equivalence (cf. [17,18]). We recall that a derived equivalence : D(X ) → D(Y ) of surfaces induces a homomorphism CH : CH * (X ) → CH * (Y ) between the numerical Chow groups. Then one says that is filtered if CH (0, 0, 1) = (0, 0, 1). In Proposition 3.1 we show that in the case of abelian surfaces, a filtered equivalence induces an isomorphism between the surfaces. Hence in order to complete the proof of Theorem 1.2, we construct an equivalence of derived categories where is an autoequivalence such that the moduli space M Y (v) of Gieseker-stable sheaves with v = CH (0, 0, 1) is a smooth surface that admits a universal family U. This completes the proof as the composition of with the Fourier-Mukai functor associated to U is a filtered equivalence. We have been able to perform this strategy only if the rank component r of CH (0, 0, 1) is non-zero. In the other case r = 0, we had to involve Mukai's equivalence S Y : D(Y ) → D( Y ) induced by the Poincaré bundle in order to apply the same plan. This explains why the conclusions of Theorem 1.2 and [6, Theorem 5.1] are not completely symmetric.
In Sect. 6 we observe that one can push the techniques of [17] a little further in order to prove that K 3 surfaces that are canonical covers of Enriques surfaces in characteristic p > 3 do not admit any non-trivial Fourier-Mukai partner. This in particular extends the second part of the result of Sosna [34, Theorem 1.1] to positive characteristic. Theorem 1.3 Let S be an Enriques surface over an algebraically closed field of characteristic p > 3 and let X be its canonical cover. Then any smooth projective surface that is derived equivalent to X is isomorphic to X .

Notation
Unless otherwise specified we work over an algebraically closed field k of positive characteristic p.

Fourier-Mukai transforms and Chow groups
Let k be an algebraically closed field of positive characteristic p. The bounded derived category of sheaves of a smooth projective variety X is defined as D(X ) := D b Coh(X ) . The category D(X ) is k-linear and triangulated. If Y is another smooth projective variety, an object E in D(X × Y ) defines a Fourier-Mukai functor via the assignment: where p 1 and p 2 denote the projections from X × Y onto the first and second factor respectively. An important theorem of Orlov tells us that any equivalence F : Finally we recall that the composition of Fourier-Mukai transforms is again of Fourier-Mukai type. Now consider an abelian surface A over k. We denote by CH * (A) num = ⊕ i CH i (A) num the graded ring of algebraic cycles modulo numerical equivalence so that Finally, we denote the Mukai pairing on CH * (A) num,Q by from (2.1), and by the fact that v • E = CH E • v. We conclude this subsection by pointing out the following peculiar fact true for abelian surfaces. Its proof is identical to that of [7,Corollary 9.43] with the opportune modifications. Moreover it holds in any dimension.

Proposition 2.1 If
: is an equivalence of derived categories of abelian surfaces, then We will tacitly use the previous result throughout the rest of the paper.

Some examples of (auto)equivalences
We denote by A an abelian surface and by A its dual variety. Moreover let P be the normalized Poincaré line bundle on A × A so that S A := P : D(A) → D( A) is an equivalence of triangulated categories [23]. The action of S CH A swaps the first and third entry of a vector, e.g.: Let now H be a line bundle on A and h be its class in CH 1 (A). The autoequivalence T A (H ⊗n ) : D(A) → D(A) (n ∈ Z) defined by F → F ⊗ H ⊗n acts on the numerical Chow rings as: Finally, the shift functor [1] : D(A) → D(A) acts on CH * (A) by − 1.

Isogenies and exponents
If A is an abelian variety over k, we denote by n A : A → A the multiplication-by-n-map and by A[n] its kernel. We say that an elliptic curve E over k is ordinary (resp. supersingular) [16] Proof Since E and F are supersingular, their j-invariants are defined over F p 2 . Moreover by [14,Corollary 78] there exists an isogeny ξ : F F p 2 → E F p 2 of degree l r for some positive integer r 0. Therefore we obtain our desired isogeny from ξ by extension of scalars. Finally we observe that ξ is separable as the degree of every non-separable isogeny is divisible by char(k) [33, Corollary 2.12].

Line bundles on a product of two elliptic curves
Let (E, O E ) be an elliptic curve over k. We denote the Mumford bundle on E × E by is the diagonal divisor and pr 1 , pr 2 are the projections of E × E onto the first and second factor respectively. Given another elliptic curve (F, O F ), line bundles L E and L F on E and F respectively, and a morphism ϕ : F → E, we define a line bundle on the product E × F where pr E and pr F are the projections onto E and F respectively.

Proposition 2.5
If ϕ : F → E and ψ : F → E are isogenies, then Proof The proof is a simple application of the see-saw principle.
If L E , L E and L F , L F are line bundles on E and F respectively such that d E : Corollary 2.6 With notation as in Proposition 2.5, in CH 1 (E × F) there are equalities of classes where d E    Proof Denote by L E the restriction of L to E × {O F }, and similarly let L F be the restriction We note that the restriction of L to {O E } × F is trivial, while the restrictions L | E×{y} lie in Pic 0 (E) for all y ∈ F. Thus by the universal property of the dual variety [26, Theorem on p. 117], there exists a unique morphism ϕ : F → E such that the conclusion of the proposition follows by setting ϕ = η −1 ϕ.

Lifting results
Let k be a perfect field of positive characteristic p and let W (k) be the ring of Witt vectors with quotient field K . We recall that it is a complete discrete valuation ring such that K is of characteristic zero (see for instance [31, Sect. II.6]). We will denote by W both W (k) and finite extensions of W (k). If X denotes a smooth projective scheme over k, a projective lift of X over W is a flat morphism ψ : X → W where X is a projective scheme, together with the choice of an isomorphism between the closed fiber X k and X . Grothendieck Finally, if ϕ : A → B is a morphism between ordinary abelian varieties, then there exists a unique morphism ϕ : Another result that we will need in the sequel is the existence of liftings of étale covers. A reference for the following theorem is [32, Sect. IX, 1.10].

Theorem 2.9
Let S be the spectrum of a complete local Noetherian ring, and let X → S be a proper S-scheme. Moreover denote by X 0 the closed fiber over the unique closed point of S. Then the assignment X → X × X X 0 yields an equivalence between the category of finite étale coverings of X and the category of finite étale coverings of X 0 .

Moduli spaces
Let A be an abelian surface defined over a field k and let h ∈ NS(A) be the class of an ample line bundle. Given a vector v = (r , l, χ) ∈ CH * (A) with integral coefficients, we consider the moduli space M h (v) of Gieseker-semistable sheaves with Mukai vector v, where stability is computed with respect to h. We want to show that, under certain assumptions, these spaces are irreducible and are Fourier-Mukai partners of A. We will need the following criterion for fully faithfulness: Now we can turn to the proof of the following: (2.5) [2]. This functor sends any stable sheaf on A whose corresponding point lies on M ⊂ M h (v) to a skyscraper sheaf at that point on M, while it sends all the stable sheaves on A that are not on M to zero (see for example [13, Proof of Theorem 2.2]-the proof uses only that there are no nontrivial homs between stable sheaves of the same slope so it works also in positive characteristic). On the other hand, for any sheaf F on M, the Mukai vector of U (F) only depends on the Mukai vector of F as it is obtained by convoluting v(F) with v(U ∨ ). This leads to a contradiction as skyscraper sheaves and the zero object have different Mukai vectors.

Observe that we have an isomorphism of functors
Finally, we only have to show that U is an equivalence. We will prove that it is fully faithful using Theorem 2.10, and then conclude by [7,Proposition 7.6]. We remark that, even though Chapter 7 of [7] is in characteristic 0, this proposition uses only the fully-faithfulness of U and some general results about k-linear categories and adjoint functors, so it works in our setting.
By [25,Proposition 3.12] the universal family U is strongly simple. In particular we have that So we just need to check that there is a skyscraper sheaf such that By the definition of universal family the functor U sends a skyscraper sheaf k(x) to a stable [2] is the left adjoint of U , the strong simplicity of U implies that Hom D(M h (v)) (E, k(x)) k, and so we are finished.

Relative moduli spaces
We also need to consider relative moduli spaces of Gieseker-semistable sheaves on a projective lift f : that is a coarse moduli space for the subfunctor of families of pure Gieseker-stable sheaves. Thus, if (r , χ) = 1 (i.e. every Gieseker-semistable sheaf is Gieseker-stable), then . Note that if we denote the closed fiber of f by A k and the geometric generic fiberby A η , then there are isomorphisms is the moduli space of pure Gieseker-stable sheaves with vector v η = (r ,l |A η , χ) on A η , and h η is the restriction of h to A η . The smoothness of the moduli space implies that f : M A/W ( v) → W is formally smooth and of finite type, and hence flat. So, with a choice of isomorphism In [17,Theorem 6.1] the authors prove that if two K 3 surfaces have a filtered equivalence between them, then they are isomorphic. The proof of this statement is quite involved and uses deformation theory of complexes in order to lift a derived equivalence of K 3 surfaces in positive characteristic to an equivalence of K 3 surfaces in characteristic zero. Here we notice that a filtered equivalence of abelian surfaces still induces an isomorphism. As the kernel of an equivalence of abelian varieties is a sheaf (up to shift), its proof turns out to be rather simple.

Proposition 3.1 Let : D(A) → D(B) be a filtered equivalence between the derived categories of two abelian surfaces. Then there exists an isomorphism f : A → B and a line bundle L on A × B such that is isomorphic to (− ⊗ L) • f * up to shift. In particular A and B are isomorphic.
Proof Equivalences of derived categories of abelian varieties send (up to shift) structure sheaves of points O x to sheaves. This is proved in [5,Lemma 10.2.6] in characteristic zero, but its proof extends to positive characteristic without any change. Hence we can suppose that (O x ) is a sheaf with Mukai vector (0, 0, 1), so it is itself a skyscraper sheaf. Since the argument holds for all points x in A, the proposition follows by [7,Corollary 5.23].
We now prove Theorem 1.2 of the introduction, which relies on the following technical proposition. (iii) r is coprime with χ.
We split our argument in several steps.  Let By (3.1), it is possible to choose an n so that χ 1 + n I is relatively prime with r 1 . Let B 1 be a line bundle with numerical class b 1 . By (2.3) the homomorphism T C (B ⊗n 1 ) • 1 CH sends Our choice of n ensures that r 2 and χ 2 are relatively prime. Step Choose d large enough so that the second component is an ample class and set CH

FM partners of canonical covers of hyperelliptic surfaces
In this and the next section, we will work, unless otherwise specified, under the assumption that the characteristic p of the base field is bigger than 3. We denote the set of Fourier-Mukai partners of a smooth projective variety X by In the case of an abelian variety A, we say that its set of Fourier-Mukai partners is trivial if FM(A) = {A, A}.
A hyperelliptic surface over an algebraically closed field k of positive characteristic p > 3 is a smooth projective minimal surface X with K X ≡ 0, b 2 (X ) = 2, and such that each fiber of the Albanese map is a smooth elliptic curve (cf. [1,Sect. 10]). These surfaces can be described as quotients (E × F)/G of two elliptic curves E and F by a finite group G. The group G acts on E by translations, and on F in a way such that F/G P 1 . Moreover, there are only a finite number of possibilities for the action of G on E × F, which have been classified by Bagnera-De Franchis [1, 10.27].
By [1,Sect. 9.3] the order n of the canonical bundle of X is finite with n = 2, 3, 4, 6. Therefore we can consider the canonical cover π : X → X of the surface X , which is the étale cyclic cover associated to the canonical bundle ω X . The degree of π is the order n of ω X , and in addition π comes equipped with an action of the cyclic group that realizes X as the quotient X /(Z/nZ). According to Bagnera-De Franchis' list [1, 10.27], the canonical cover X of an arbitrary hyperelliptic surface X = (E × F)/G is an abelian surface that sits inside a tower of surfaces where π is an étale cyclic cover of degree at most three. Moreover, if π has degree three, then F admits an automorphism group of order three and has j-invariant equals to zero. Therefore the dual morphism π realizes the dual of X as one of the following varieties: (i) the product E × F, (ii) an étale cyclic cover of E × F of degree two, or (iii) an étale cyclic cover of E × F of degree three such that F has an automorphism group of order three.

The work of Sosna
In [34, Theorem 1.1] the author proves that the set of Fourier-Mukai partners of the canonical cover of a complex hyperelliptic surface is trivial. By using Bagnera-De Franchis' classification, Sosna's theorem boils down to proving the following result concerning derived equivalences of special abelian surfaces.

Theorem 4.1 (Sosna) Let E and F be complex elliptic curves and let A be a complex abelian surface. Then the set FM(E × F) is trivial. Moreover, if E × F → A is a degree two étale cyclic cover, then FM(A) is trivial. Finally, the same conclusion holds if E × F → A is a degree three étale cyclic cover and rk NS(A) ∈ {2, 4}.
In view of Theorem 2.9 we prefer to work with étale covers rather than quotients. Thus we reformulate Sosna's theorem in the following version.

Proposition 4.2 Given a complex abelian surface A, then the set FM(A) is trivial in the following cases: (i) A is isomorphic to the product E × F of two elliptic curves; (ii) A is a degree two étale cyclic cover of a product E × F of two elliptic curves; (iii) A is a degree three étale cyclic cover of a product E × F of two elliptic curves and rk NS(A) ∈ {2, 4}.
Proof If A → E × F is a cover of degree one, two, or three, then the dual isogeny E × F → A realizes A as a quotient of two elliptic curves. Then by Theorem 4.1 we conclude that FM( A) is trivial. As FM(A) = FM( A) and rk NS(A) = rk NS( A), the proposition follows at once.
As an application of Proposition 4.2, we deduce some further finitiness results that will be useful towards the proof of Theorem 1.1. Proof We show that the dual abelian variety A satisfies the hypotheses of Proposition 4.2. The result will follow as FM(A) = FM( A). Let q be either 2 or 3 and consider an isogeny ψ : E × F → A of exponent q such that ψ •ϕ = q A (recall that q A denotes the multiplicationby-q-map on A). As deg q A = q 4 and deg ϕ = q 3 , we deduce that deg ψ = q. Hence the dual isogeny ψ is a cyclic cover of E × F of order q. The second statement follows as rk NS(A) = rk NS( A).

Strategy of the proof of Theorem 1.1
Since an abelian surface and its dual have the same Fourier-Mukai partners, the following theorem implies Theorem 1.1.

Theorem 4.4 Let A be an abelian surface over an algebraically closed field of characteristic p > 0. Then FM(A) is trivial in the following cases:
(i) A is isomorphic to the product E × F of two elliptic curves; (ii) p > 2 and A is a degree two étale cyclic cover over the product E × F of two elliptic curves; (iii) p > 3 and A is a degree three étale cyclic cover over the product E × F of two elliptic curves such that F admits an automorphism of order 3.
In order to prove the previous theorem, we will consider the following set of hypotheses Setting 4. 5 We denote by E and F two elliptic curves over an algebraically closed field k of characteristic p > 0. Moreover we set ν : A → E × F to be either an isomorphism of abelian surfaces, or an étale cyclic cover of degree d ν = 2, 3 (as in the hypotheses of Theorem 4.4). Finally we assume that p > deg ν.

Remark 4.6
Since the exponent of an isogeny divides its degree, the exponent of the isogeny ν of Setting 4.5 is either 1 if ν is an isomorphism, or d ν otherwise. Now let μ : E × F → A be an isogeny of exponent d ν such that μ•ν = (d ν ) A . Then the dual isogeny μ : A → E × F is either an isomorphism, or else its degree and exponent satisfy (deg μ, exp μ) = (d 3 ν , d ν ).
As an application of Theorem 2.9 we deduce that both the isogenies ν and μ of Setting 4.5 and Remark 4.6 lift to the ring of Witt vectors. In the following result we check that their degrees and exponents remain unchanged when passing from the special fiber to the general fiber.

Proposition 4.7 Let E and F be elliptic curves and ϕ : A → E × F be an étale isogeny of abelian surfaces with degree relatively prime with p. If E → R and F → R are projective lifts of E and F over an Henselian local ring R with residue field k, then there exists a projective lift A → R of A and an isogeny ϕ R :
A → E × R F such that ϕ R lifts ϕ and its restriction ϕ η : A η → E η × F η to the geometric general fibers is an isogeny with deg ϕ η = deg ϕ and exp ϕ η = exp ϕ Proof By Theorem 2.9 there is a projective lift A → R of A and an étale cover ϕ R : A → E × R F that specializes to ϕ. Up to composing ϕ R with a translation of E × R F , we can suppose that ϕ R is a homomorphism of groups. We are going to prove that the restriction of ϕ R to the geometric generic fiber of A is an isogeny ϕ η : A η → E η × F η such that deg ϕ η = deg ϕ. To see this we notice that the kernel K of ϕ R is a finite étale group over R and moreover, as ϕ is separable, we have where K k is the closed fiber and K η is the geometric generic fiber. In addition, since K is a finite étale group scheme over R, the closed fiber is killed by multiplication by n if, and only if, the same is true for the geometric generic fiber. So we have also equality of the exponents.
We will deduce Theorem 4.4 from the following technical proposition. satisfies the following conditions: Set λ = ν if C = A, and λ = μ otherwise (see Remark 4.6). In addition assume that there exist projective lifts E → W and F → W of E and F over a finite ramified extension of the ring of Witt vectors respectively such that the following conditions hold: Then B is either isomorphic to C or C. In particular we have that FM(A) is trivial.
Proof We divide our argument in several steps.
Step 1. We first prove that there exists a projective lift B → W of B such that the geometric generic fiber B η is derived equivalent to C η . Let L be an ample line bundle on C with class l. Then Theorem 3.3 implies that B is isomorphic to a moduli space M l (v) of Gieseker-stable shaves with Mukai vector v = (r , l, χ) ∈ CH * (C). Now consider a preimage L of L under ρ as in (A 1 ) and the relative moduli space where v l := (r , l, χ) and l is the class of L. As discussed in Sect. 2.7, this is a projective lift of B and the geometric generic fiber M η is a moduli space of Gieseker-stable sheaves on C η with Mukai vector v η = (r , l |C η , χ). Therefore as discussed in Theorem 2.11, the conditions (E1), (E3), and v η , v η C η = 0 imply that M η is an abelian surface. In addition there exists a universal family U η on M η × C η that induces an equivalence U : D(C η ) → D(M η ).
Step 2. Now we prove that under the assumptions of Theorem 4.4 the abelian surface C η is isomorphic to either M η or its dual M η . By the Lefschetz principle we can suppose that the abelian surface C is defined over a subfield of the complex numbers C and therefore that C η is defined over C. Suppose first that ν : A → E × F is an isomorphism. Then both λ and λ W are isomorphisms and therefore so is the restriction λ η : C η → E η × F η of λ W to the geometric generic fibers. As a product of elliptic curves has no non-trivial Fourier-Mukai partners (Theorem 4.1), there is an isomorphism C η M η . Suppose now that ν : A → E × F is a degree two cyclic cover. By Remark 4.6 we have (deg λ, exp λ) ∈ {(2, 2), (8, 2)} and by Proposition 4.7 we have (deg λ η , exp λ η ) = (deg λ, exp λ). Therefore by Propositions 4.2 and 4.3 we deduce that either C η M η or C η M η . The case when ν has degree three follows similarly by using the condition (A 2 ).
Step 3. The argument of [17, Lemma 6.5] (based on a result of Matsusaka-Mumford) proves that the isomorphism C η M η (resp. C η M η ) between the geometric generic fibers of the two liftings induces an isomorphism C B (resp. C B) between the closed fibers. This immediately yields that either B A or B A, and hence that FM(A) is trivial.

Finding a suitable equivalence
In this section we finish the proof of Theorem 1.1. According to Proposition 4.8, we only need to verify its hypotheses. We work under the hypotheses of Setting 4.5 and assume that the abelian surface B is a Fourier-Mukai partner of A. In the following we will examine two cases: (a) at least one of the two elliptic curves E or F is ordinary, and (b) both E and F are supersingular.
The satisfies one the two following statements: (i) r is relatively prime with both p 1 and p 2 ; (ii) either p 1 divides r but not χ and p 2 divides χ but not r , or vice versa.

Proof
Set v 0 := (r 0 , l 0 , χ 0 ) = CH (0, 0, 1) Since is an equivalence, by (2.2) we have that Let H 0 be a line bundle on A such that its class in the Néron-Severi group is h 0 . Therefore at the level of numerical Chow rings the equivalence n := We divide the proof in five cases.
Case I Suppose that neither p 1 nor p 2 divides χ 0 . In this case the equivalence is given by the composition S A • : D(B) → D( A).
Case II Suppose that both p 1 and p 2 divide both r 0 and χ 0 . By (5.1) we see that I is relatively prime with p 1 and p 2 as well. Now choose a positive integer n coprime with both p 1 and p 2 . Therefore by looking at the definition (5.2) of χ n , this immediately implies that χ n is relatively prime to both p 1 and p 2 . We conclude then as in Case I.
Case III Suppose that both p 1 and p 2 divide r 0 , and that precisely one of them, say p 1 , divides χ 0 . We choose a positive integer n such that n is relatively prime to both p 1 and p 2 . By (5.1) I is relatively prime to p 1 , and by (5.2) p 1 does not divide χ n . Moreover, again by (5.2) and the fact that n is general, we can suppose that p 2 does not divide χ n as well. We then set := S A • n .
Case IV Suppose that both p 1 and p 2 divide χ 0 and that precisely one of them, say p 1 , divides r 0 . In this case we proceed as in Case III by considering the composition S A • in place of .
Case V Suppose that one of the primes, say p 1 , divides both r 0 and χ 0 , but p 2 divides neither r 0 nor χ 0 . Let n = p 2 and consider n . By (5.1) p 1 does not divide I , and hence p 1 does not divide χ n . Moreover, by our choice of n, we have that p 2 does not divide χ n as well. We then conclude as in Case I. satisfies one of the two following conditions: (a) r 0 is relatively prime to both p 1 and p 2 , or (b) one of the primes p 1 and p 2 divides r 0 but not χ 0 , while the other divides χ 0 but not r 0 . Set λ = ν if C = A and λ = μ otherwise. We claim that there exists an equivalence 1 : D(B) → D(C) such that in the vector v 1 := (r 1 , λ * l 1 , χ 1 ) = CH 1 (0, 0, 1), r 1 is positive and relatively prime with p, and l 1 ∈ CH 1 (E × F). In order to prove the claim, we distinguish the two cases (a) and (b) mentioned above. First consider case (a). As r 0 is not zero, we can make it positive by composing with the shift functor, if necessary. Let L be a line bundle representing l 0 and m be a positive integer such that p 2 2 divides (r 0 m + 1). Hence we can write 1 + r 0 m = p 2 2 u for some integer u and we consider the composition T C (L ⊗m ) • . Then for some integer χ 1 . By Proposition 2.3 there exists a class l 1 ∈ CH 1 (E × F) such that ( p 2 2 u) l 0 = λ * (u l 1 ). This proves the claim in case (a) as we can set 1 := T C (L ⊗m ) • and v 1 := (r 0 , (p 2 2 u) l 0 , χ 1 ). Now consider case (b). If necessary we replace with S C • in order to make r 0 divisible by p 1 = p but relatively prime with p 2 = deg ν. Let L and m be as above. Then the equivalence 1 := T C (L ⊗m ) • sends the vector (0, 0, 1) to CH 1 (0, 0, 1) = r 0 , p 2 2 u l 0 , χ 0 + m l 2 0 + r 0 m 2 l 2 where l 2 is a class in CH 1 (C). Since v 0 , v 0 C = 0, we have that l 2 0 = 2r 0 χ 0 and hence As p 1 divides r 0 but does not divide χ 0 , it follows that Hence, for some elements ω ∈ CH 1 (C) and ω ∈ CH 1 (E × F). Also, p 2 = deg ν does not divide r 0 (here we use the fact that S CH C induces an homomorphism CH 1 (C) → CH 1 ( C); see Sect. 2.2). If necessary we can make the integer χ 2 positive by composing with the shift functor. This concludes the proof of the claim. Now let w = (s, h, ξ) be the Mukai vector of CH 1 (1, 0, 0) so that w, v 1 C = 1. Moreover let H be a line bundle whose numerical class is h and note that Choose a positive integer n such that n p 2 2 I + χ 1 ≡ 0 (mod q) for every prime divisor q = p 2 of r 1 that does not divide I . Now set 2 := T C (H ⊗(n p 2 2 ) )• 1 and (r 2 , l 2 , χ 2 ) := CH 2 (0, 0, 1). With a simple calculation we find We note that r 2 is not divisible by p, and moreover that by (5.4), χ 2 is not divisible by any prime divisor q = p 2 of r 1 that does not divide I . On the other hand, if a prime divisor q = p 2 divides both r 1 and I , then by (5.3) it does not divide χ 1 , and hence does not divide χ 2 . Finally we prove that χ 2 is not divisible by p 2 in the case where p 2 divides r 1 . But this follows by the construction of 1 , and by noting that in the case (b) discussed earlier, p 2 does not divide r 0 . Now consider the composition 3 0 is a positive integer. By a direct computation we have that CH 3 sends the vector (0, 0, 1) to where θ is the numerical class of and By Proposition 2.3 and (5.4) we can write l 2 = λ * l 3 for some class l 3 ∈ CH 1 (E × F), and hence as the isogeny λ has degree either p c 2 with c = 1 if C = A, and c = 3 otherwise. Therefore the first component of v 3 is still positive and relatively prime with p, while the second component is a pull-back of an ample class in CH 1 (E × F) (for d 0). Moreover χ 3 ≡ χ 2 (mod r 2 ) and hence it is still relatively prime with r 2 = r 1 . Our desired equivalence is hence given by Now we consider separately the case of supersingular abelian surfaces and the case of abelian surfaces that are not supersingular.

The case where one of the two curves is ordinary
The above results imply that when one of the elliptic curve involved is not supersingular then we can construct an equivalence satisfying (E 1 ), (E 2 ) and (E 3 ). In order to conclude we need to prove that the hypotheses (A 1 ) and (A 2 ) of Proposition 4.8 hold as well. We will first prove a couple of auxiliary results.
Proof As Hom(E, F) = 0 we obtain an isomorphism Pic(E × F) Pic(E) × Pic(F). Let now pr * E L E ⊗ pr * F L F be an arbitrary line bundle on E × F. Since line bundles on curves lift, we can consider lifts L E and L F of L E and L F respectively. Hence the line bundle  Proposition 4.7. It follows that λ * W L lifts λ * L which proves condition (A 1 ). In order to prove (A 2 ) we can assume that F has an automorphism of order three. Then by Theorem 2.8 the geometric generic fiber F η admits an automorphism of order three and hence rk CH 1 (E η × F η ) ∈ {2, 4} where E η is the geometric generic fiber of the lift E → W . and (E 3 ) of Proposition 4.8. By Proposition 5.4 we can choose general lifts E and F of E and F to W respectively and a line bundle L ∈ Pic(C) such that the condition (A 1 ) holds (see the argument of Proposition 5.5). Furthermore, since any fixed elliptic curve has a 1-dimensional family of deformations and for any choice of E, there are only finitely many elliptic curves isogenous to E η , we may choose E and F so that there are no non-trivial morphisms between the geometric generic fibers E η and F η . It follows that rk NS(E η × F η ) = 2 independently of whether F admits an automorphism group of order three or not. As in Proposition 5.5 this immediately implies that rk NS(C η ) = 2.

Supersingular case
We are going to prove that the hypotheses of Proposition 4.8 hold when the elliptic curves E and F are both supersingular. The following proposition proves the conditions (E 1 ), (E 2 ) and (E 3 ). As (r 1 p n)ψ is either the zero morphism or non-separable, we have that γ + (r 1 p n )ψ is a separable isogeny. Now choose a positive integer n so that p p 2 2 n I + χ 2 ≡ 0 (mod q) for every prime divisor q = p 2 of r 2 that does not divide I .
By the same argument as Proposition 5.2, this choice of n ensures that the third component of v 3 is relatively prime with r 1 . Now we define the line bundle : on E × F and we consider the equivalence T C (λ * ⊗(r 1 d) ) • 3 with d 0 a positive integer. Take θ := l(0 E , 1, 1) to be the class of . The equivalence T C (λ * ⊗(r 1 d) ) • 3 sends (0, 0, 1) to v 4 = r 1 , λ * (γ + r 1 n p ψ, m 1 + r 1 p n k 1 + r 2 1 d, m 2 + r 2 p n k 2 + r 2 1 d), χ 4 where χ 4 := χ 2 + p p 2 2 n I + r 1 p 2 p 4 2 n 2 h 2 2 As χ 4 is congruent to the third component of v 3 modulo r 1 , we have that χ 4 is still relatively prime with r 1 . Moreover, for d sufficiently large the second component of v 4 is ample. Finally, r 1 is relatively prime with p, and hence T C (λ * ⊗(r 1 d) )• 3 is the equivalence we are looking for.
To conclude the proof we need to analyze the case when ϕ is separable. But in this case the proof is simpler as there is no need to introduce the isogeny ξ and the equivalence 2 . Then it is enough to set γ = ϕ and proceed as in the inseparable case.   [29, p. 189 or p. 172]) it is possible to lift a supersingular elliptic curve together with an endomorphism over a finite ramified extension W of the ring of Witt vectors. Thus we have lifts F 0 → W and ρ W : F 0 → F 0 of F and ρ to W . The étale cover ϕ : E → F determines a lift E 0 of E together with a lift ϕ 0 of ϕ over W . Now, denote by λ W ,0 : C 0 → E 0 × W F 0 the lift of λ as defined in Proposition 4.8. After noting that there is an isomorphism (1 E × ϕ) * M E ( ϕ × 1 F ) * M F , and hence that the previous argument shows that the line bundle λ * L(ϕ, M 1 , M 2 ) lifts to C 0 . Moreover the restriction of ρ W to the geometric generic fiber (F 0 ) η is an automorphism that is not proportional to the identity. Hence rk CH 1 (E 0 ) η × (F 0 ) η is either 2 or 4 according to whether or not there is an isogeny between (E 0 ) η and (F 0 ) η . As the rank of the Néron-Severi group does not change under separable isogenies, we have that rk CH 1 ((C 0 ) η ) ∈ {2, 4} where (C 0 ) η is the geometric generic fiber of C 0 . In particular the assumption (A 2 ) holds.

Canonical covers of Enriques surfaces
An Enriques surface S over an algebraically closed field k of characteristic p = 2 is a smooth projective minimal surface with canonical bundle ω S of order two, χ(O S ) = 1, and b 2 (S) = 10. The canonical cover π : X → S induced by ω S is a double étale cover with X a K 3 surface. Let : D(Y ) → D(X ) be a derived equivalence so that Y is itself a K 3 surface by [17,Proposition 3.9]. We aim to prove Theorem 1.3, namely that X Y .
In [17] the authors prove that Shioda-supersingular K 3 surfaces do not admit any nontrivial Fourier-Mukai partners, thus we can assume that X has Picard rank less than 22. Since in odd characteristic Shioda-supersingularity is equivalent to Artin-supersingularity (cf. for instance [30]), we can assume that the formal Brauer group of X is of finite height. In particular there exists a lift X → W of X over the ring of Witt vectors such that NS(X ) NS(X η ), where as usual X η is the geometric generic fiber [28, p. 505]. By the work of Jang [12,Theorem 2.5] there is a primitive embedding of the Enriques lattice (2) := U (2) ⊕ E 8 (−2) of S into NS(X ) such that the orthogonal complement of the embedding does not contain any vector of self intersection −2. Using the isomorphism NS(X ) NS(X η ), together with the characterization of Enriques-K 3 surfaces in terms of its periods [27, Theorem 1.14], we deduce that X η is a K 3 surface arising as the canonical cover of an Enriques surface. In particular X η does not have any non-trivial Fourier-Mukai partner thanks to [34, = (r , l, χ). Since all the line bundles on X deform to line bundles over X → W , we can consider the relative moduli space M X /W (r , l, χ) → W where l is a lift of l. The geometric generic fiber of M X /W (r , l, χ) → W is by construction a Fourier-Mukai partner of X η and therefore, by the previous considerations, it is isomorphic to X η . We conclude by using [17,Lemma 6.5], which says that the isomorphism between the geometric generic fibers of relative K 3 surfaces induces an isomorphism between the closed fibers. thank M. Olsson for useful advices and P. Sosna for carefully reading a first draft of this paper. Moreover KH thanks A. Bertram and E. Canton for helpful conversation; LL thanks A. Bellardini, C. Liedtke, C. Schnell and M. Talpo for comments and correspondence; ST thanks A. Bertram, C. Hacon, M. Lieblich, B. Moonen, K. Schwede, M. Talpo and B. Viray for very interesting mathematical conversation. She was also partially supported by the grant 261756 of the Research Councils of Norway. Finally the authors thank their previous institutions, where part of these project were ultimated, that respectively were: University of California at Berkeley and University of Utah, University of Bonn, Stonybrook University, and University of Florence, and the University of Utah. LL was partly supported by the SFB/TR45 "Periods, moduli spaces, and arithmetic of algebraic varieties" of the DFG (German Research Foundation). All authors are grateful to the unknown referee for carefully reading the paper, and for suggesting corrections to the discussion in Sect. 2.7 and a more direct proof (as well as the current more general statement) of Proposition 4.7.
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