A refined Derived Torelli Theorem for Enriques surfaces

We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from $2$ are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof of a conjecture by Ingalls and Kuznetsov relating the derived categories of the blow-up of Artin-Mumford quartic double solids and of the associated Enriques surfaces. This paper originated from one of the problem sections at the workshop `Semiorthogonal decompositions, stability conditions and sheaves of categories', Universit\'e de Toulouse, May 2-5, 2018.


Introduction
An Enriques surface is a smooth projective surface X with 2-torsion dualizing sheaf ω X and such that H 1 (X, O X ) = 0. In this paper we assume that all varieties are defined over an algebraically closed field K of characteristic different from 2. Under this additional assumption, the above definition is equivalent to asking that X is the quotient of a K3 surface by a fixedpoint-free involution.
In characteristic zero, Enriques and K3 surfaces share one important Hodge theoretic feature: their period maps are injective. In other words, Hodge-theoretic Torelli theorems hold for Enriques and K3 surfaces. On the other hand, the bounded derived category of coherent sheaves D b (X) has very different behaviour depending on whether X is an Enriques or a K3 surface. Indeed, in the first case, D b (X) uniquely determines X up to isomorphism (see [8]) while if X is a K3 surface, there might be numerous (albeit finitely many) K3 surfaces with derived category equivalent to D b (X) (see [31,34]).
From the derived category point of view, the same picture is still true in positive characteristic, under our assumptions on the field K, in view of [16,29]. We will refer to the combination of these results as the Derived Torelli Theorem for Enriques surfaces: two Enriques surfaces X 1 and X 2 , defined over a field K as above, are isomorphic if and only if D b (X 1 ) ∼ = D b (X 2 ).
If X is a K3 surface, then D b (X) is indecomposable, i.e. it does not contain proper non-trivial admissible subcategories. On the other hand, we will recall in Section 3.1 that, for a generic Enriques surface X, we have a semiorthogonal decomposition (1) D b (X) = Ku(X, L), L , with L := {L 1 , . . . , L 10 } an orthogonal exceptional collection.
We will refer to the admissible subcategory Ku(X, L) as the Kuznetsov component of X.
This decomposition and the properties of Ku(X, L) have been extensively studied in [19,24]. The aim of this paper is to investigate further how much of the geometry of X is encoded by Ku(X, L). The following result should be thought of as a refined version of the Derived Torelli Theorem for Enriques surfaces mentioned above.
Let us remark that this statement contains the non-trivial implication of the more general Theorem 5.1 which is proved in Section 5.1.
The terminology used above will be clarified in Section 2 while the proof of this result will be carried out in Section 5. The essence of the proof in the non-trivial direction is in proving that the equivalence F can be extended to an equivalence D b (X 1 ) ∼ = D b (X 2 ) (see Proposition 2.6). Then the Derived Torelli Theorem implies our result. The way we obtain the latter global equivalence is by studying and classifying the so called 3-spherical objects in Ku(X i , L i ). This is done in Section 4. Note that the reason why we have to assume that the characteristic of K is not 2 is because the Derived Torelli Theorem for Enriques surfaces is known to hold only under this additional assumption, but the rest of our argument applies even in characteristic 2 for 'classical Enriques surfaces' (see [13]).
The techniques used to prove the theorem above can be adapted to give a short proof of a very interesting conjecture by Ingalls and Kuznetsov (see Conjecture 3.5). This will be explained in detail in Section 3.2 and here we content ourselves with a short summary. Let us now work over K = C and consider the blow-up Y ′ of an Artin-Mumford quartic double solid at its 10 singular points. It is a classical observation that Y ′ has an associated Enriques surface X obtained as a quotient of the (desingularization) of the ramification locus. The main result of [19] shows that there is a semiorthogonal decomposition where {G 1 , . . . , G 10 } is an orthogonal exceptional collection. Most importantly, there is an exact equivalence Ku(Y ′ ) ∼ = Ku(X, L) of Fourier-Mukai type (see Theorem 3.4).
The following is our second main result.
Theorem B. In the situation above, there is an exact equivalence This statement, which is precisely the content of the conjecture of Ingalls and Kuznetsov, was previously proved by Hosono and Takagi [17] using an intricate argument depending on Homological Projective Duality. Our more precise Theorem 5.3 in Section 5.2 implies Theorem B and, given its elementary proof, it provides a simpler proof of such a conjecture. This paper originates from the circle of ideas that stems out of [5]. Indeed, if Y 1 and Y 2 are cubic threefolds (i.e. smooth degree 3 hypersurfaces in the complex projective space P 4 ) and H i is a hyperplane section of Y i , then we have semiorthogonal decompositions The main result in [5] shows that Y 1 ∼ = Y 2 if and only if there is an exact equivalence Ku(Y 1 ) ∼ = Ku(Y 2 ). To make the analogy with the paper [5] tighter, we should also mention cubic threefolds and quartic double solids are both Fano threefolds of index 2 and their derived categories admit very similar semiorthogonal decompositions (see Corollary 3.5 in [19]), even though quartic double solids are in general singular.
If we increase the dimension of the hypersurfaces by one and we consider two cubic fourfolds W 1 and W 2 , then we get semiorthogonal decompositions similar to the one above but with additional exceptional objects O W i (2H i ). In this case, the admissible subcategories Ku(W i ) behave like the derived category of a K3 surface. Thus, in view of the discussion above, we cannot expect that W 1 ∼ = W 2 if and only if there is an exact equivalence Ku(W 1 ) ∼ = Ku(W 2 ). And in fact, Huybrechts and Rennemo proved in [18] that such a statement needs an adjustment: the equivalence Ku(W 1 ) ∼ = Ku(W 2 ) has to satisfy some additional and natural compatibility. Contrary to the approach we use in the present paper, the strategies in [5], the appendix to [3], and [28] all make use of Bridgeland stability conditions.
In conclusion, it is worth pointing out that arbitrary semiorthogonal decompositions are in general non-canonical. However, Theorem A is further evidence of the fact that, when these decompositions originate from geometry, they usually encode important pieces of information.

Semiorthogonal decompositions and an extension result
In this section, we briefly recall some basic facts about semiorthogonal decompositions. We also prove and extension result for Fourier-Mukai functors of independent interest and which will be important in this paper.
2.1. Generalities. In complete generality, let T be a triangulated category. A semiorthogonal decomposition is a sequence of full triangulated subcategories D 1 , . . . , D m of T such that: (a) Hom(F, G) = 0, for all F ∈ D i , G ∈ D j and i > j; (b) For any F ∈ D, there is a sequence of morphisms The subcategories D i are called the components of the decomposition. The condition (a) implies that the factors C i (F ) in (b) are uniquely determined and functorial. Hence, for all i = 1, . . . , m, one can define the i-th projection functor π i : T → D i such that Denote by ι i : D i ֒→ T the inclusion. We say that D i is admissible if ι i has left adjoint ι * i and right adjoint ι ! i . Let T 1 = D 1 1 , D 1 2 and T 2 = D 2 1 , D 2 2 be two traingulated subcategories with semiorthogonal decompositions by admissible subcategories ι ij : D j i ֒→ T j . Following [26, Section 2.2], we can define the gluing functor 1 The following result will be useful later.
2 be a triangulated category with a semiorthogonal decomposition by admissible subcategories. Let F : T 1 → T 2 be an exact functor with left and right adjoints and such that 1 Note that our gluing functor differs from the one in [26] by the shift by 1. This is harmless and it makes the rest of the discussion easier.
(a) F j (A) ∈ D j 2 and F j is an equivalence, for j = 1, 2 and for all A ∈ D j 1 .
for all A ∈ D 1 1 and all B ∈ D 1 2 . Then F is an equivalence.
Proof. The objects of D 1 1 and D 1 2 form a spanning class Ω for T 1 in the sense of [7]. Hence, by [7,Theorem 2.3], to prove that F is fully faithful it is enough to show that, for any A, B ∈ Ω, the natural morphism induced by F is bijective.
Since F j is an equivalence by (a), we just need to verify that the morphism induced by F is bijective, for all A ∈ D 1 1 and B ∈ D 1 2 . To this extent, consider the commutative diagram where the top and bottom rows are obtained by applying F. Note that the vertical arrows are isomorphism by adjunction and (b). Since F 1 is an equivalence by (a), the top row is an isomorphism. Thus the bottom one is an isomorphism as well.
The essential surjectivity can be deduced now by a standard argument. Indeed, observe that every object in T 2 is isomorphic to the cone of a morphism f : A → B, where A ∈ D 2 1 and B ∈ D 2 2 . By our assumption (a) and the fully faithfulness of F, we can lift f : A → B to a morphismf :Ā →B in T 1 . Hence the cone of f is isomorphic to the image under F of the cone off and F is essentially surjective.
In the general situation where we have a semiorthogonal decomposition then, π 1 and π 2 coincide with the left adjoint ι * 1 and and the right adjoint ι ! 2 of the embeddings ι i : D i ֒→ T . Hence we can define a functor From now on, let us assume that all categories are linear over a field K. An object E ∈ T is exceptional if Hom(E, E[p]) = 0, for all integers p = 0, and Hom(E, E) ∼ = K. A set of objects {E 1 , . . . , E m } in T is an exceptional collection if E i is an exceptional object, for all i, and Hom(E i , E j [p]) = 0, for all p and all i > j. An exceptional collection {E 1 , . . . , E m } is orthogonal if Hom(E i , E j [p]) = 0, for all i, j = 1, . . . , m with i = j and for all integers p.
If T admits a semiorthogonal decomposition T = D 1 , D 2 with D 2 = E 1 , . . . , E m , where {E 1 , . . . , E m } is an orthogonal exceptional collection, then the left mutation through D 2 takes a particularly convenient form:

2.2.
Extending Fourier-Mukai functors. Now let X 1 and X 2 be smooth projective varieties over K with semiorthogonal decompositions Remark 2.2. Under the assumption that X is a smooth projective variety, all the components in a semiorthogonal decomposition of D b (X) are admissible (see [6]).
Denote by ι ij : D j i ֒→ D b (X j ) the embedding. Assume further that where E i is an exceptional object. Denote by ζ i : A i ֒→ D i 1 the embedding. Since, all subcategories in the above decompositions are admissible, they are endowed with a Serre functor S C , where C is any of the admissible subcategories above (see, for example, [19, Lemma 2.8]).
Proof. We have a distinguished triangle On the other hand, we have a distinguished triangle where the morphism on the right is the unique (up to scalar) non-trivial morphism given by Serre duality. By definition S i ∈ A i .
Since S i ∈ A i and Hom(A, C i ) ∼ = 0, for all A ∈ A i , the role of the two distinguished triangles in the argument above can be exchanged and so there are unique k ′ 1 : As at the beginning of this section, assume that we have semiorthogonal decompositions Remark 2.4. It should be noted that, when F is an equivalence, [25, Conjecture 3.7] would imply that F ′ is of Fourier-Mukai type. Unfortunately, this conjecture is not known to hold true in the generality needed in this paper. This expectation should be compared with the fact that any full functor F : [10,32]).
As in the setting of Lemma 2.3, assume further that D i 1 := A i , E i , where E i is an exceptional object. Let F : A 1 → A 2 be an exact functor of Fourier-Mukai type and denote by Φ E : D b (X 1 ) → D b (X 2 ) the corresponding Fourier-Mukai functor. The following result provides a criterion for the extension of Φ E . Lemma 2.5. In the assumptions above, if F : 2 is of Fourier-Mukai type and is trivial on S X 1 (D 1 2 ). We will only use the functor Φ E • π D 1 1 , and for the convenience of notations, we still denote the kernel by E.
Therefore, we get a natural morphism which is obtained by composing the above isomorphism with the counit morphism For notational simplicity, in the rest of the argument we remove the embeddings ι 1j .
By adjunction, we have Note that by flat base change and the projection formula we have The fact that ΦẼ | S X 1 (D 1 2 ) is trivial follows directly from the definition of Φ E andẼ. By composing with the projection functor π ′ By combining Lemmas 2.1 and 2.5 we get the following useful result.
Proposition 2.6. In the assumptions of Lemma 2.5, if F : A 1 → A 2 is an exact equivalence, then there exists a Fourier-Mukai functor ΦẼ : Proof. The existence ofẼ and (1) follow directly from Lemma 2.5. In particular, since ΦẼ | D 1 2 is trivial, to prove (2) we can first assume that D i 2 is trivial, for i = 1, 2. At this point, we want to apply Lemma 2.
Since (1) implies hypothesis (a) in Lemma 2.1, it remains to check that ΦẼ satisfies (b) in the same lemma. To this extent, note that, given any A i ∈ A i , if we apply RHom(ζ i (A i ), −) to the distinguished triangle in Lemma 2.3, then by Serre duality, the composition by η E i induces an isomorphism for i = 1, 2. Hence, by functoriality and the assumption on F, for any A ∈ A 1 , we get the following commutative diagram where Φ and Ψ are obtained by applying ΦẼ. We just need to show that the vertical arrow on the right in (8) is an isomorphism. By using the definition ofẼ (see, in particular, (5)) and the distinguished triangle (3) in Lemma 2.3, we have the following commutative diagram: If we analyze the construction of the morphism (4) in the proof of Lemma 2.5, it is easy to see that the bottom row is given by the non-trivial extension as such a row gets canonically identified with the extension provided by the morphism defining E. Thus ΦẼ (η E 1 ) sits in a distinguished triangle The same argument as in the proof of (7) yields that the rightmost vertical arrow in (8) is an isomorphism, as ΦẼ (A) ∈ A 2 by (1), for all A ∈ A 1 .

2.3.
A dg category apprach. Even though our approach in this paper is purely triangulated, this section ends with a short discussion concerning an alternative viewpoint via dg categories (the non-expert reader can have a look at [9] for a quick introduction to this subject). Let us first rediscuss semiorthogonal decompositions in terms of dg categories. Assume that T = D 1 , D 2 is an algebraic triangulated category with ι i : D i ֒→ T an admissible subcategory, for i = 1, 2. This means that there is a pre-triangulated dg category C and an exact equivalence T ∼ = H 0 (C), where H 0 (C) denotes the homotopy category of C. In particular D 1 and D 2 are algebraic, in the sense that there exist two pre-triangulated dg subcategories I i : As it is explained for example in [26,Section 4], there exists a pre-triangulated dg category C 1 × ϕ C 2 and an isomorphism C ∼ = C 1 × ϕ C 2 in Ho(dgCat). Here Ho(dgCat) denotes the homotopy category of the category dgCat of (small) dg categories which are linear over the field K. As it is suggested by the notation, the definition of C 1 × ϕ C 2 depends on the choice of a dg bimodule ϕ (i.e. a C • 1 ⊗ C 2 -dg module). For this reason, we will refer to it as the gluing of C 1 and C 2 along ϕ. By [26,Corollary 4.5], the dg bimodule ϕ yields, at the trianglated level, the gluing functor in Section 2.1.
Assume that C 1 and C 2 are two pre-triangulated dg categories which are obtained by this gluing procedure. In explicit form, Example 2.7. For our applications, we should think of the case Ku( where X i is an Enriques surface and L i is a suitably chosen exceptional object in L i . Moreover, we assume that C i j are taken so that Ku( . From this, it is easy to guess how to choose C i and C i j to deal with the setting in Section 5.2. Let J 1 : C 1 1 → C 2 1 and J 2 : C 1 2 → C 2 2 be two isomorphisms in Ho(dgCat).
Remark 2.8. In our geometric setting, the condition that the exact equivalence F : Ku(X 1 , L 1 ) → Ku(X 2 , L 2 ) is of Fourier-Mukai type is equivalent to requiring that F lifts to an isomorphism in Ho(dgCat) between the natural dg enhancements of the semiorthogonal components induced by those of D b (X i ) (see, for example, [9, Section 6.3]).
At this point, we would like to conclude that J 1 and J 2 glue to an isomorphism J : C 1 → C 2 in Ho(dgCat). To this extent, consider the (C 1 1 ) • ⊗ C 1 2 -module ϕ ′ 1 obtained from ϕ 2 by composing with J 1 and J 2 . Suppose that there is a quasi-isomorphism between ϕ 1 and ϕ ′ 1 . Roughly, this means that ϕ 1 and ϕ 2 are compatible under the action of J 1 and J 2 . At the triangulated level, this corresponds to the assumption on F in Lemma 2.5.
The following is a well-known result, which should be thought of as the dg analogue of Proposition 2.6. Proposition 2.9. In the assumptions above, there is an isomorphism J : C 1 → C 2 in Ho(dgCat) whose restriction to C 1 j is J j , for j = 1, 2. Proof. The argument is an easy adaptation of the proofs of Propositions 4.11 and 4.14 in [26].
In general, the condition to check in order to glue J 1 and J 2 to get J is of dg nature and not easy to verify. In the special case where C i 2 is generated by an exceptional object, as in Section 2.2, Lemma 2.5 shows that the verification is actually of triangulated nature.
It should be noted that this approach to semiorthogonal decompositions was put in an even more general categorical setting in [33] (see, in particular, [33, Section 9.3]).

The geometric setting
In this section we discuss the structure of the derived categories of Enriques surfaces and of Artin-Mamford quartic double solids. The emphasis is on the analogies which are at the core of this paper.
To the best of our knowledge, the result above is not known when K has characteristic 2, due to the additional complexity of the double cover structure.
As we mentioned in the introduction, the category D b (X) has a very nice description in terms of semiorthogonal decompositions when X is generic. To be more precise, we begin by recalling that a Fano polarization ∆ ∈ Pic(X) is an ample divisor (with numerical class δ) satisfying (a) δ 2 = 10; (b) |δ.f | ≥ 3 for every 0 = f ∈ Num(X) with f 2 = 0. By [13, Corollary 2.5.5], δ defines a unique isotropic 10-sequence, that is, a set of 10 isotropic vectors {f 1 , . . . , f 10 } ⊂ Num(X) := NS(X)/NS(X) tor such that f i .f j = 1 − δ ij and δ = 1 3 (f 1 + · · · + f 10 ). Conversely, given such an isotropic 10-sequence {f 1 , . . . , f 10 }, the numerical class δ satisfies δ 2 = 10 and δ.f i = 3 for all i = 1, . . . , 10. In particular, if all of the f i 's are nef numerical classes, e.g. if X is generic in moduli, then δ is the numerical class of a Fano polarization. In such a case, for each i = 1, . . . , 10 we may choose divisors , and it was observed by Zube in [35] that if K = C then the set of line bundles {O X (−F + 1 ), . . . , O X (−F + 10 )} forms an orthogonal exceptional collection. We should also observe that generalizations of Theorem 3.1 are available in the twisted setting (see [1] when the characteristic is 0 and [16] in positive characteristic).
For sake of completeness, we discuss and prove the following well-known result that explains when we can find such an orthogonal exceptional collection. Recall that an Enriques surface X is called unnodal if it contains no smooth rational curves. It is called nodal otherwise. Nodal Enriques surfaces form a divisor in their moduli space.
Proof. We begin by observing that it suffices to show that we may choose the f i so that ∆ = a contradiction. The same argument shows that Hom(O X (−F + i ), O X (−F + j + K X )) = 0 for i = j. Therefore, it follows that for i = j we have we get the orthogonality, as claimed. As mentioned above, for generic X, or more generally X unnodal, we may choose an arbitrary effective isotropic 10-sequence (for this, use [15], [14] or [30, Corollary 3.12]). Then ∆ as above will automatically be ample. If X is a generic nodal Enriques surface, then it admits a Reye polarization [15], a special Fano polarization in which the embedding of X ⊂ P 5 is contained in a smooth quadric (see [14,Section 5]). Choosing the corresponding isotropic 10-sequence gives the result in this case.
It follows from the lemma that, up to codimension 2 in moduli, an Enriques surface X has a semiorthogonal decomposition as in (1) For an extensive discussion about some remarkable properties of Ku(X, L) for nodal Enriques surfaces, one can have a look at [19].
. . , R i n i −1 is a chain of (−2)-curves in a tree.

Artin-Mumford quartic double solids.
Let us now assume that K = C and consider two vector spaces V and W of dimension 4. Consider the divisor Clearly, Q s can be thought of as a family of quadrics in P(V ) parametrizied by P(W ). The degeneration locus of this family of quadrics is a (singular) quartic surface D s ⊆ P(W ) which is usually called the quartic symmetroid. For s generic, D s has 10 singular points which correspond to quadrics with corank 2.
On the other hand, we can consider the (singular) double covering Y s → P(W ) ramified along D s . When s is generic, we will refer to Y s as the Artin-Mumford quartic double solid since it was explained in [12] that they are precisely the Artin-Mumford conic bundles constructed in [2] as examples of unirational but not rational conic bundles. From now on, for simplicity, we will remove the section s from the notation. Now, given an Artin-Mumford quartic double solid Y , the blow-up of the ramification quartic symmetroid D at its 10 singular points is a K3 surface D ′ with an involution ι acting without fixed points. Indeed, the surface D ′ can be seen as the zero locus of the section s seen as a global section of W ∨ ⊗ O P(V )×P(V ) (1, 1) and ι is just induced by the transposition of factors P(V ) → P(V ). The quotient X := D ′ /ι is then an Enriques surface which will be called the Enriques surface associated to Y . These Enriques surfaces are nodal.
Let us now discuss the homological side of this picture. Let Y be an Artin-Mumford quartic double solid and let Y ′ be its blow-up at the 10 singular points. By [19,Lemma 3.6], the variety Y ′ is the double covering of the blow-up Z of P(W ) at the corresponding 10 points ramified along the proper transform D ′ of D.
By [19,Lemma 3.12], we have a semiorthogonal decomposition subcategories where h is the class of the hyperplane in P(W ) and e i is the class coming from the i-th exceptional divisor in the blow-up of P(W ) at the image of the 10 singular points in Y . We will not need this but one can be more explicit as and denote by S A Y ′ its Serre functor. In [19], the authors exhibit an embedding of X inside the Grasmannian Gr(2, V ) which provides an ample polarization of degree 10 on X. Then, by Lemma 3.2, the Enriques surface X associated to Y has an explicit semiorthogonal decomposition The following statement collects the main results in [19] concerning A Y ′ and Ku(Y ′ ).
This result strongly suggests that A Y ′ should be very much related to the derived category of an Enriques surface. In other words, it is very suggestive to guess that the correspondence between Artin-Mumford quartic double solids and associated Enriques surfaces might have a nice categorical counterpart. In fact, this was made precise by the following [19, Conjecture 4.2]: Conjecture 3.5 (Ingalls-Kuznetsov). If Y ′ is the blow-up of an Artin-Mumford quartic double solid Y at its 10 singular points, then there is an equivalence A Y ′ ∼ = D b (X), where X is the Enriques surface associated to Y . This conjecture was proved by Hosono and Takagi in [17]. As we explained in the introduction, one of the aims of this paper is to provide another short and simple proof (see Theorem B and Theorem 5.3 in Section 5.2 for a more precise statement).

Spherical objects in Enriques categories
The proof of Theorem A is based on the classification of some spherical objects in Ku(X, L), for X an Enriques surface. We will explain this in a slightly more general setting which is suited to deal with the case of Artin-Mumford quartic double solids as well.
Recall that if T is a K-linear triangulated category with Serre functor S T and d is a positive integer, we have the following.
In particular, the graded Ext-algebra of a d-spherical object is isomorphic to the cohomology of a sphere of dimension d.
We are interested in studying this kind of objects in categories which resemble the derived category of an Enriques surface. We now consider the following very general situation:  Other examples are provided by Gushel-Mukai threefolds and fourfolds (see [27,Proposition 4.5]). Roughly speaking, these manifolds are smooth n-dimensional intersections of the cone in P 10 over Grassmannian Gr(2, 5) ֒→ P 9 with P n+4 ⊆ P 10 and a quadric hypersurface Q ⊆ P n+4 . We omit here all the details as these varieties do not play a role in this paper.
In the generality of Setup 4.3, pick any exceptional object M i in A Z . To simplify the notation we set S := S A Z . By Serre duality, we have Hom(M i , S(M i )) ∼ = K. Hence we can consider the cone S i of the non-zero morphism An easy exercise using Serre duality, the formula for L M 1 ,...,M N from (2), and the orthogonality of the M i 's shows that there is a natural isomorphism and thus S i ∈ Ku(Z).
The following is an easy exercise using the definition of S i . Proof. By using for example [23, Lemma 2.6], we can easily compute To conclude the proof that the objects S i are (2m − 1)-spherical, we apply the functor RHom(S(M i ), −) to the distinguished triangle (10). This yields Hence, to prove that S i is (2m − 1)-spherical, it is enough to apply RHom(−, S i ) once more to (10) and observe that S i ∈ Ku(Z). The last part of the proof follows from an easy diagram chasing using (10)   The lemma shows how to construct (2m − 1)-spherical objects in Ku(Z). We want to show now that the list is complete. if we set N := 1≤i≤N M i , we get the following distinguished triangle (11) F → N ⊗ RHom(N, S(F )) → S(F ).
If we apply RHom(F, −) to it, then, by Serre duality and taking into account that F is (2m−1)spherical, we get a distinguished triangle of graded K-vector spaces Hence there is an integer k and an isomorphism and so there are i, j = 1, . . . , N such that To conclude that i = j, we may apply RHom(−, M j ) to (11) to get an instant contradiction as RHom(F, Hence, up to reordering, we may assume i = 1. And up to shifting we can further assume To simplify the notation, for the rest of the proof, we set M := M 1 . Therefore, (12) gets the following simplified form Now, let C be the cone of the unique (up to scalars) morphism It follows that C, F and M sit in a distinguished triangle coming from the commutative diagram:  (15)) and using the fact that F ∈ Ku(Z), we see that Serre duality allows us to conclude that non-trivial morphisms f and g exist. It remains to show that, up to scalar, f •g and g•f are the identity. If we apply RHom(−, S(F )) to (14), we deduce that the map By combining this with the previous arguments, we have the commutative (up to scalars) diagram of non-trivial morphisms Thus the composition of any two subsequent maps in the top row is non-trivial. Therefore, For the rest of the paper, we will actually need only the following simple consequence. Remark 4.9. It would be interesting to study whether Proposition 4.7 and Corollary 4.8 can be generalized to situations where one has semiorthogonal decompositions as in Remark 3.3. Unfortunately, the proofs provided above do not apply in this setting.

Proofs of the main results
The proofs of the main results in this paper follow the same line of argument and consist in extending the equivalences between Kuznetsov components. We will carefully explain this for the proof of Theorem A, and we will briefly outline how this can be adapted to prove Theorem B as well.
5.1. Proof of Theorem A. We start with two semiorthogonal decompositions where L i := L i 1 , . . . , L i 10 and satisfies (1). We are also given an exact equivalence with an isomorphism of exact functors ι 12 • F • ι * 11 ∼ = Φ E . We just need to show that Φ E induces an equivalence Ψ : as Theorem 3.1 would then allow us to conclude that We begin by considering for any j = 1, 2 and any i = 1, . . . , 10, the 3-spherical object S j i ∈ Ku(X j , L j ) defined by the distinguished triangle By Corollary 4.8, we have F(S 1 i ) = S 2 i , up to reordering and possibly shifting. By Remark 4.6 (ii), the assumptions of Proposition 2.6 are satistfied for for n = 1, 2. Thus we get an equivalence F 1 : D 1 1 → D 2 1 of Fourier-Mukai type with Fourier-Mukai functor ΦẼ 1 . The argument proceeds inductively. To simplify the notation slightly, let us set L i := L 1 i and M i := L 2 i . For k ≥ 2, we assume that we have a Fourier-Mukai kernelẼ k−1 such that where we setẼ 0 := E. To proceed to stage k, we apply again Proposition 2.6 with for n = 1, 2. Here it is important to stress that, since the exceptional objects in L n are orthogonal, by Remark 4.6 (ii) the object ζ n ζ ! n (L n k ) ∼ = S n k is actually contained in Ku(X n , L n ) and not just in A n . Hence ΦẼ k−1 satisfy the assumptions of Proposition 2.6 by induction. Continuing in this way, we get an equivalence ΦẼ 10 : D b (X 1 ) ∼ − → D b (X 2 ) so that X 1 ∼ = X 2 , as claimed.
As we mentioned in the introduction, we can reverse the implication in the statement of Theorem A. Indeed, if there exists an isomorphism f : X 1 → X 2 , then f * : D b (X 2 ) → D b (X 1 ) is a Fourier-Mukai equivalence. Take on D b (X 2 ) the semiorthogonal decomposition of type (1) described in Section 3.1. The equivalence f * yields an analogous one on D b (X 1 ) with an exact equivalence F : Ku(X 2 , L 2 ) 1i . Since Φ i is a Fourier-Mukai functor by [22,Theorem 7.1], the functor F ′ is of Fourier-Mukai type being isomorphic to a composition of Fourier-Mukai functors.
Combining the arguments in this section we get the following more general version of Theorem A.
Remark 5.2. One way to generalize Theorem 5.1 to non-generic Enriques surfaces could be to consider semiorthogonal decompositions as in Remark 3.3. As it was suggested by A. Perry, one might then try to deform X 1 , X 2 and the equivalence F : Ku(X 1 , L 1 ) → Ku(X 2 , L 2 ) to the generic case. Unfortunately, while in the generic case first order deformations of X i coincide with first order deformations of the subcategory Ku(X i , L), this seems not to be the case for nodal non-generic Enriques surfaces. Thus one needs to add some natural assumptions on F. Namely we need the equivalence to preserve commutative first order deformations of X 1 and X 2 . This will be investigated in future work.

5.2.
Proof of Theorem B. The proof of our second main result is now easy. Indeed, let Y ′ be the blow-up at the 10 singular points of an Artin-Mumford quartic double solid Y with associated Enriques surface X. Consider the semiorthogonal decompositions , . . . , O X (−F + 10 ) discussed in Section 3.2. Moreover, we know that there is an exact equivalence Ku(X, L) ∼ = Ku(Y ′ ) of Fourier-Mukai type (see Theorem 3.4 (ii)). By Corollary 4.8, the 3-spherical objects in Ku(Y ′ ) are all obtained (up to shifts and isomorphism) via the construction in Section 4. The extension procedure described in Section 5.1 then applies verbatim.
In particular, we get the following more precise version of Theorem B. (2) Up to reordering and shifts, Φ E (L i ) = G i , for i = 1, . . . , 10.
In a sense, this result proves a stronger version of Conjecture 3.5 since the equivalence we construct is automatically compatible with the semiorthogonal decompositions of D b (X) and A Y .
Acknowledgements. This work began in a problem-solving working group as part of the Workshop on "Semiorthogonal decompositions, stability conditions and sheaves of categories" held at the University of Toulouse in 2018. It is our pleasure to thank this institution and the organizers of the workshop for the very stimulating atmosphere. Marcello Bernardara, Daniele Faenzi, Sukhendu Mehrotra, and