Birational geometry of moduli spaces of stable objects on Enriques surfaces

Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.


Introduction
Moduli spaces of stable sheaves on surfaces are much studied objects. As stability depends on the choice of a polarization, it is interesting to study the dependence of the geometry of the moduli spaces on this choice. The introduction of Bridgeland stability conditions [5] prompted new techniques, which can be applied to study this question. In [2,3], Bayer and Macrì have analyzed in detail the birational geometry of moduli spaces on a projective K3 surface X. In particular, they proved that crossing a wall induces a birational transformation and that every smooth K-trivial birational model of a moduli space can be obtained by varying stability conditions in the distinguished connected component Stab † (X) of the stability manifold discovered by Bridgeland [6,Def. 11.4].
The purpose of this paper is to prove analogous results for moduli spaces of stable objects on an Enriques surface Y . The main technique is to consider the covering K3 surface Y and use the already established results for Y .
Using the pullback along π, we get a 2:1 morphism of the moduli space M Y σ (v) onto a Lagrangian subvariety of M Y σ (π * (v)) [11,18]. Applying a result by Marian and Zhao [15], we conclude Proposition 1.1 (see Proposition 4.2). Let v ∈ H * alg (Y, Z) be a Mukai vector such that π * (v) ∈ H * alg ( Y , Z) is primitive and σ ∈ Stab † (Y ) a generic stability condition. The image of the morphism is a constant cycle Lagrangian.
Recall that a subvariety is called constant cycle if all its points become rationally equivalent in the ambient variety.
It turns out that this cycle-theoretic property is enough to deduce birational equivalence of moduli spaces with different stability conditions.
If the rank is even, we deduce that the moduli spaces are Calabi-Yau manifolds employing results by Saccà [20]. To obtain the results for not necessarily generic Enriques surfaces one uses deformation theory and stability conditions in families [1].
As a final application of the birational equivalence of wall-crossing we consider the natural nef divisor classes ℓ σ associated to a stability condition σ [3,Sec. 4]. Since the birational transformation of Theorem 1.2 can be obtained as the restriction of the birational transformation of the moduli spaces on the K3 surface, we can furthermore prove that these divisors can be extended continuously to a map see Lemma 4.11. We conclude the paper by showing that the nef and semiample divisors ) are big (Proposition 4.12) and discuss whether all minimal models of moduli spaces of stable objects on an Enriques surface are again a moduli space for a (possibly) different stability condition.
Relation to other work. The independent preprint [19] deals as well with the birational geometry of moduli spaces of stable objects on Enriques surfaces. It is shown in loc. cit. that on an Enriques surface and for arbitrary Mukai vector v with v 2 > 0 the moduli spaces M σ (v) and M τ (v) are birational, where σ and τ are generic stability conditions. This is established, in analogy to [2,Thm. 5.7], by classifying all possible birational phenomena that occur on a wall for v, see [19,Thm. 5.8] for the precise result. Although these results are more general than ours and yield a detailed analysis on all possible wall-crossing types, the approach in this paper is of independent interest and enables one to prove the birational equivalence in important cases in only a few pages. Via deformation theory we can obtain most of the results of [ Notations and conventions. We work over the complex numbers. The bounded derived category of a smooth projective variety X is denoted by D b (X). Throughout, Chow groups are the groups of cycles modulo rational equivalence. An Enriques surface is called generic if the Picard rank of its covering K3 surface is 10. In Section 4, unless otherwise specified, we only consider generic Enriques surfaces. The moduli spaces on Enriques surfaces under consideration have two components and we frequently omit the determinant in the notation.

Constant Cycle Subvarieties
Our approach uses the following class of subvarieties. Observe that if X is symplectic, then every constant cycle subvariety is isotropic, which follows from Roitman's Theorem [23,Prop. 10.24]. Thus, in the definition one may replace the Lagrangian assumption with a condition on the dimensions 2 dim(Y ) = dim(X).
We refer to [9,24] for further discussions on constant cycle subvarieties. For our purposes we only need the following properties. Proof. Consider the commutative diagram where ι denotes the inclusion of both subvarieties. If X is smooth, the birational map f * induces an isomorphism between CH 0 (X) and CH 0 (Y ) and the assertion follows from the commutativity of the diagram. For arbitrary X, use a resolution of singularities π : X → X and argue as above using the above diagram and the corresponding one for π. ✷ The following result is the main ingredient for our proof of Theorem 1.  If for very general x ∈ Z there exists a fiber p −1 2 (a) with x ∈ p 1 (p −1 2 (a)) and a rational curve x ∈ C x ⊂ p 1 (p −1 2 (a)) such that C x is contained in Z, then Z would be uniruled, in contradiction to kod(Z) ≥ 0.
Hence, we may assume that for very general x ∈ Z the rational curves C x ⊂ p 1 (p −1 2 (a)) through x are not contained inside Z. However, then the constant cycle subvariety T strictly contains Z and, thus, is of larger dimension. Since the dimension of a constant cycle subvariety inside a symplectic variety is bounded by half the dimension of the ambient space, we derive a contradiction. and for each φ ∈ R a full additive subcategory P(φ) ⊂ D satisfying the following conditions • Every 0 = E ∈ D has a categorical Harder-Narasimhan filtration.
• Z factors through the numerical Grothendieck group K num (D).
• There exists a constant C > 0 such that for all φ ∈ R and objects E ∈ P(φ) we Objects in P(φ) are semistable of phase φ and simple objects in P(φ) are called stable.

Bridgeland constructed on the set of stability conditions Stab(D) a generalised metric to
show that it is a complex manifold [5, Thm. 1.2]. We only consider stability conditions for the bounded derived category D b (X) of a smooth projective surface X.
The space of stability conditions comes naturally with two group actions. The first one is the group of exact autoequivalences Aut(D b (X)) acting on the left via ψ.
Similiar to the case of Gieseker stability, the space of stability conditions Stab(X) admits For K3 surfaces X, the Mukai lattice is contained in H * (X, Z). For Enriques surface, since td(X) = (1, 0, 1 2 ), we have to allow rational coefficients in H 4 alg (X, Z). We first review briefly moduli spaces of stable sheaves on K3 surfaces. A Mukai vector v = (r, c 1 , s) is positive if r > 0, or r = 0, c 1 is effective and s = 0, or r = c 1 = 0 and s > 0. Bridgeland constructed certain geometric stability conditions on K3 surfaces and showed that they all lie in a distinguished component Stab † (X) of the stability manifold [6]. For stability conditions in Stab † (X), Bayer and Macrì showed that the above theorem also holds true for moduli spaces of stable complexes [3]. Another key result by the two authors, which we need in our investigation of the birational type of moduli spaces, is the following, cf. [2].
Now we pass to Enriques surfaces Y . Since their canonical divisor ω Y is 2-torsion, the moduli space for the Mukai vector v decomposes into where we furthermore fix the determinant line bundle L, respectively L ⊗ ω Y . If the rank of the Mukai vector is odd, the two components are isomorphic.
Kim [11,12] was the first one to study moduli spaces of stable sheaves on Enriques surfaces.
Proposition 3.4 (Kim). Given an Enriques surfaces Y with its universal cover π : Y → Y , the morphism has degree 2 and it is étale onto its image away from all points image is the fixed locus of the action given by the covering involution i * ∈ Aut(M Y π * H (π * (v))) and is a Lagrangian subvariety.
Successive results by Yoshioka [27], Hauzer [8] and finally by Nuer [17,18] have lead to a complete understanding of when the moduli spaces are non-empty. We only need The category of coherent sheaves Coh(Y ) on an Enriques surface Y is naturally isomorphic to the category of coherent i * -sheaves Coh i * ( Y ). This yields a natural equivalence between the bounded derived categories D b (Y ) and D b i * ( Y ). In [14], the authors described stability conditions on Enriques surfaces using i * -invariant stability conditions on Stab † ( Y ) via the functors π * and π * . For a generic Enriques surface Y we denote by σ ∈ Stab † ( Y ) the stability condition Nuer [18] established the existence of projective coarse moduli spaces for Bridgeland stability conditions on Enriques surfaces. For primitive Mukai vector these are also smooth projective K-trivial varieties and the morphism ) is 2:1 onto the fixed locus of the action given by the covering involution.

Moduli Spaces of stable objects on Enriques Surfaces
We first observe that the image of the moduli space of stable objects on an Enriques surface is a constant cycle Lagrangian. Shen, Yin, and Zhao [22] studied the group of zerocycles on moduli spaces of stable objects on K3 surfaces. They formulated a conjecture, which was later proven by the third author and Marian [15].
By [4], Enriques surfaces Y satisfy Bloch's conjecture. Since their Albanese variety is trivial, we get CH 0 (Y ) = Z. Thus, we conclude the following for all Enriques surfaces Y Observe that this argument does not solely work for Enriques surfaces. For example, one may take a K3 surface X given as a 2:1 cover X → P 2 . Similarly, one may consider K3 surfaces with a non-symplectic automorphism of finite order. The quotient also satisfies Bloch's conjecture.
The moduli space M Y σ (v) itself is not always CH 0 -trivial. Indeed, these moduli spaces have Kodaira dimension zero which implies that the one-dimensional moduli spaces are elliptic curves. However, as we will see later, moduli spaces parametrizing odd rank Mukai vectors are always CH 0 -trivial.

Wall-crossing for generic Enriques surfaces. Recall that we have an action of
. By the work of Macrì, Mehrotra, and Stellari [14] we know that the action of the covering involution i * is trivial on Stab † ( Y ), since Y is generic. This yields and we concentrate only on one component, omitting the determinant in the notation.
The above lemma enables us to show that spherical twists are equivariant functors with respect to the covering involution. Recall that a Fourier-Mukai functor Φ ∈ Aut(D b ( Y )) is

Proof. The sperical twist is the autoequivalence with Fourier-Mukai kernel
Consider now two stability conditions σ + , σ − ∈ Stab † (Y ). Inside one chamber the moduli spaces M Y σ+ (v) and M Y σ− (v) stay the same. Hence, we only need to study the relationship of these two moduli spaces for σ + and σ − in adjacent chambers. We can assume that the corresponding stability conditions σ + and σ − in Stab † ( Y ) are also generic with respect to π * (v) and lie in adjacent chambers since Y is generic. Proof. To ease notation, we prove the statement at first only for odd rank Mukai vectors and describe at the end how to deduce the result in the even rank case. Recall that under is an embedding of a constant cycle Lagrangian since we only consider one component.
Consider the moduli spaces M Y σ+ (v) and M Y σ− (v) in adjacent chambers and embed them inside M Y σ+ (π * (v)) and M Y σ− (π * (v)) respectively. We know the assertion for the two moduli spaces of stable objects on the K3 surface Y . For our purpose we need an additional property of the birational map. Observe that, since i * acts on the moduli spaces M Y σ+ (π * (v)) and M Y σ− (π * (v)), it makes sense to ask whether the birational map is i * -equivariant.
The occuring birational map depends on the wall in the sense of [2,Thm. 5.7]. There are three different types. The first type induces a divisorial contraction of the moduli space.
The contraction map contracts curves of stable objects that become S-equivalent for a stability condition on the wall. The second type is a wall inducing a flopping contraction.
The remaining case is a fake wall, i.e. there are no curves in M Y σ+ (π * (v)) and M Y σ− (π * (v)) that become S-equivalent with respect to a stability condition on the wall.
We now treat each of these cases and show that the corresponding birational map is equivariant.
In case of a flopping contraction or a fake wall there either exists a common open subset whose complement has at least codimension two or the birational map is induced by the composition of spherical twists. Using Lemma 4.4 we see that in both cases the map is equivariant.
The case of a wall inducing a divisorial contraction is divided into three subcases. If we are in the Brill-Noether case, the birational map is again defined on an open subset to be a sequence of spherical twists associated to stable spherical objects.
The second type is the Hilbert-Chow case. Here, the proof uses an isotropic vector The last occurring type is called Li-Gieseker-Uhlenbeck. We again have a moduli space can be identified with the fixed set of the involution i * and f is equivariant, the restriction to the constant cycle Lagrangian gives the desired birational transformation. This finishes the proof for Mukai vectors of odd rank.
If the rank is even, we can still deduce that the image of each of the two components of the moduli space M Y σ ± (v) under π * intersects the open set U of the map f . Indeed, the pullback morphism π * is étale onto its image and, therefore, its image is still of Kodaira dimension 0. Thus, the above argument gives us an i * -equivariant Fourier-Mukai transform inducing on an open subset a birational map between π * (M Y σ+ (v)) and π * (M Y σ− (v)). This means that each of the two components of M Y σ+ (v) gets mapped to precisely one of the two components of M Y σ− (v). To deduce the result for the moduli space we just observe that π * corresponds to forgetting the equivariant structure. Indeed, the functor Φ descends to a Fourier-Mukai transform of the Enriques surface compatible with Φ as in [7,Thm. 4.5]. Thus, they give rise to the following commutative square The fact that g is birational allows us to conclude that g ′ is birational which finishes the proof. ✷

Birational moduli spaces.
We go on to study the geometry of moduli spaces of stable objects on K3 and Enriques surfaces. The strategy in both cases will be the same.
Suppose we are given Mukai vector v, which satisfies the assumption of [2, Thm. 1.1] for K3 surfaces and the ones from Theorem 4.5 for Enriques surfaces. We already know that for different generic stability conditions σ, τ the corresponding moduli spaces M σ (v) and M τ (v) are birational. Applying an autoequivalence Φ of the surface induces an isomorphism where Φ H is the corresponding cohomological Fourier-Mukai functor. Thus, moduli spaces of stable objects with respect to a generic stability condition in the same orbit as v under the action of the group of autoequivalences on the Mukai lattice are as well birational. In this way we can reduce the study of birational types of moduli spaces to the question of the orbit of the Mukai vector v in the Mukai lattice. We will use this strategy throughout this section.
Let us start with the hyperkähler manifold. Since the torsion-free part of the second cohomology of an Enriques surface is isometric to The action of P H E on cohomology is given by the reflection along the hyperplane perpendicular to v(E). Its existence is due to the recent preprint [1], where the concept of stability conditions in families is established.
All fibers of both families are smooth K-trivial varieties. By the above we know for very general b ∈ B that the fibers φ −1 (b) and ψ −1 (b) are birationally equivalent. Fix such a birational isomorphism and the closure of its graph in the product Hilb There is at least one irreducible component of the relative Hilbert scheme that contains uncountably many of these graphs. Bayer and Macrì constructed nef divisors ℓ σ ∈ NS(M Y σ ( v)) naturally associated to a stability condition σ ∈ Stab † ( Y ) [3,Sec. 4]. They can be defined using the composition In our arguments we use the following compatibility result.

Lemma 4.10. The diagram
Proof. For two adjacent chambers C + , C − we pick again stability conditions σ ± ∈ C ± and a stability condition σ 0 on the wall W separating the two chambers. We identify NS(M Y σ+ (v)) and NS(M Y σ− (v)) using the birational isomorphism f obtained from the proof of Theorem 4.5. The proof shows that these fit into the following commutative diagram Proof. Consider the images of the moduli spaces π * (M Y σ ± (v)) ⊂ M Y σ ± (π * (v)). A curve C ⊂ M Y σ ± (v) gets contracted to a point under the morphism associated to the linear system |ℓ σ 0 ,± | if and only if its image π * (C) ⊂ M Y σ ± (π * (v)) gets contracted to a point.
There are three possible cases for the contraction on the K3 side [3,Thm. 1.4]. If the wall is a fake wall, then no curve gets contracted and the same remains true on the Enriques side.
If the morphism induced by the big line bundle has an exceptional locus B that is contracted, there are two possibilities. Firstly, the induced morphism is a divisorial contraction. In this case the divisor B has negative square with respect to the Beauville-Bogomolov form and is uniruled [25,Thm. 1.2]. Secondly, the codimension of the subvariety B that is contracted is greater than one. In this case the general fiber of the restriction of π σ ± to B is even rationally chain connected and, thus, B is uniruled [25,Thm. 1.2].
In either case the image π * (M Y σ ± (v)) cannot be contained in the uniruled variety B since it is a constant cycle Lagrangian of non-negative Kodaira dimension. ✷ We now discuss minimal models of moduli spaces M σ (v) of stable objects on generic Enriques surfaces for generic v.
In the case of a K3 surface and moduli of stable sheaves, Yoshioka [26] has proven that this morphism induces an isometry between v ⊥ with the Mukai pairing and the Néron-Severi group with the Beauville-Bogomolov form on the hyperkähler variety. This result has been generalized to Bridgeland stability conditions by Bayer and Macrì [3, Thm. 6.10].
Thus, for a ∈ v ⊥ to be mapped to a big and movable divisor its square has to be positive. On the other hand, it is important to note that Nuer and Yoshioka [19,Ex. 8.23] found examples of walls, which induce a small contraction, but the moduli spaces in adjacent chambers are isomorphic. It is unclear whether or not the birational model obtained by flopping the contraction is again a moduli space.