On the Borisov-Nuer conjecture and the image of the Enriques-to-K3 map

We discuss the Borisov-Nuer conjecture in connection with the canonical maps from the moduli spaces $\mathcal M_{En,h}^a$of polarized Enriques surfaces with fixed polarization type $h$ to the moduli space $\mathcal F_g$ of polarized $K3$ surfaces of genus $g$ with $g=h^2+1$, and we exhibit a naturally defined locus $\Sigma_g\subset\mathcal F_g$. One direct consequence of the Borisov-Nuer conjecture is that $\Sigma_g$ would be contained in a particular Noether-Lefschetz divisor in $\mathcal F_g$, which we call the Borisov-Nuer divisor and we denote by $\mathcal{BN}_g$. In this short note, we prove that $\Sigma_g\cap\mathcal{BN}_g$ is non-empty whenever $(g-1)$ is divisible by $4$. To this end, we construct polarized Enriques surfaces $(Y, H_Y)$, with $H_Y^2$ divisible by $4$, which verify the conjecture. In particular, the conjecture holds also for any element $\mathcal M_{En,h}^a$, if $h^2$ is divisible by $4$ and $h$ is the same type of polarization.


INTRODUCTION
Let be an Enriques surface over ℂ, that is, a smooth projective surface with ( ) = ( ) = 0 and 2 =  . The universal covering of is given by an étale double cover map ∶ → where is a 3 surface. Hence, an Enriques surface determines a pair ( , ) , where is its 3 cover, and is a fixed-point-free involution on so that coincides with the quotient map → ∕ . In particular, studying Enriques surfaces is equivalent to studying pairs ( , ) of 3 surfaces and fixed-point-free involutions on . A polarized Enriques surface is a pair ( , ) , where is an Enriques surface and ∈ Pic( ) is an ample line bundle. A numerically polarized Enriques surface is a pair This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Recall that Num( ) ≅ ⊕ 8 (−1) for arbitrary Enriques surface , realized by an isometry (usually called marking) ∶ Num( ) = 2 ( , ℤ) → ⊕ 8 (−1). Let ℎ ∈ ⊕ 8 (−1) be a primitive element of positive degree ℎ 2 > 0. Thanks to lattice theory, Gritsenko and Hulek were able to give a construction of the moduli space  ,ℎ of numerically polarized Enriques surfaces for some marking on , as an open subvariety of a modular variety  ,ℎ . Indeed, its points are in 1 ∶ 1 correspondence with the isomorphism classes of numerically polarized Enriques surfaces [9,Theorem 3.2]. Since this correspondence does not depend on the choice of a marking sending to ℎ, we may choose and fix a marking for each , and simply write ℎ = [ ] instead of ([ ]) . We refer to [9] for the construction and more details. The moduli space  ,ℎ is a 10-dimensional quasi-projective variety, and the locus  ,ℎ corresponding to unnodal surfaces (i.e., with no smooth (−2)-curves) is open. For an alternate approach to moduli spaces , using the invariant , parametrizing pairs of smooth Enriques surfaces and ample numerical classes [ ] ∈ Num( ) with 2 = 2 − 2 and ( ) ∶= min { . | 2 = 0, > 0 } = , we refer to [7]. Let us consider  the moduli space of polarized 3 surfaces of genus = ℎ 2 + 1. Note that is odd and ≥ 5. For any ample numerical class ℎ, we have a natural map Then the locus which appear as pullbacks of polarized Enriques surfaces ( , ) . Notice that, for any fixed degree − 1, there are only finitely many numerical classes ℎ ∈ Num( ) with ℎ 2 = − 1. Indeed, from [7,Proposition 4.16] it follows that the number of irreducible components of the moduli space ′ , coincides with the number of possible simple decomposition types for ℎ for fixed values ℎ 2 = 2 ′ − 2 and (ℎ) = . Since 0 < 2 ≤ ℎ 2 by [8, Corollary 2.7.1], there are only finitely many possible choices of , which implies the claim. Alternatively, we can argue as follows. (This was brought to our attention by A. Knutsen.) The space  ,ℎ is a quotient of similarly defined spaces of polarized Enriques surfaces, which exist by the theory of Viehweg as quasi-projective varieties, and therefore have finitely many components.
In this note, we discuss a conjecture of Borisov and Nuer on the Enriques lattice Num( ) ≅ ⊕ 8 (−1), motivated by the Ulrich bundle existence problem, and connect it to the maps ℎ . Let us briefly recall what are Ulrich bundles. Let ⊂ ℙ be a smooth projective variety of dimension , and let =  (1) be a very ample line bundle on . A vector bundle  on which satisfies the following cohomology vanishing condition is called an Ulrich bundle on (see [6]). They have many interesting applications, in particular, they connect several different topics in algebra and geometry, see [3,6]. One important problem within this topic is to find an Ulrich bundle of smallest possible rank on a given variety. For an Enriques surface , together with a very ample line bundle =  (1), it is known that always carries an Ulrich bundle of rank 2 (see [2,4]). On the other hand, Borisov and Nuer observed that the existence of an Ulrich line bundle on a polarized unnodal Enriques surface ( , ) is equivalent to the numerical condition that is, can be written as a difference of two (−2)-line bundles. Here, the unnodal assumption is required only to assure the vanishing of certain cohomology groups. Thus, it is natural to focus only on Equation (1.3). They conjectured that it is always possible to find such a line bundle for any choice of polarization , or even more, for any line bundle:  [11]). In the case, the pushforward * splits as a direct sum of two line bundles ⊕ ( ⊗ ) , where ≅ * . We consider the sublocus Since the locus Ξ is contained in the Borisov-Nuer divisor  by definition, this conjecture admits the following much weaker version: At the moment, the Borisov-Nuer conjecture is known for only a few examples: Fano polarization Δ and its multiple Δ by Borisov and Nuer themselves [5,Theorem 2.4], and a degree 4 polarization [1, Theorem 13]. In particular, Ξ is nonempty when = 5 or = 11. To have a better understanding, it is worthwhile to observe Ξ , and to collect more evidences for the Borisov-Nuer conjecture.
In from the moduli space  ,ℎ which makes Conjecture 1.1 hold. The key ingredient is a Jacobian Kummer surface = ( ) of a general curve of genus 2, similar as in [1]. Such a Jacobian Kummer surface has plenty of technical merits, for instance: • has a fixed-point-free involution , that is, is the 3 cover of some Enriques surface ; • intersection theory of is well-understood; • the pullback homomorphism * ∶ Pic( ) → Pic( ) is well-understood; • the Picard number ( ) is quite big, so there are more chances to find a certain line bundle.
The main result of this paper is the nonemptiness of the locus Ξ for various values as follows, see Theorem 3.7: Theorem 1.4. When − 1 is divisible by 4, the locus Ξ is nonempty. In other words, for any given > 0 and any Enriques surface , there is an ample and globally generated line bundle and a line bundle on such that 2 = 4 and The outline of the paper is the following. In Section 2, we review some basic facts on Enriques surfaces, Jacobian Kummer surfaces as 3 covers of Enriques surfaces, and line bundles. We also fix the notation we use. In Section 3, we describe a construction of a polarized Enriques surface which verifies the Borisov-Nuer conjecture using a Jacobian Kummer surface and we provide a few more examples in the case when ( − 1) is not divisible by 4.

PRELIMINARIES
We recall some basic facts on Enriques surfaces and Jacobian Kummer surfaces. As the above discussion indicates, we translate the Borisov-Nuer conjecture and Equation (1.3) on an Enriques surface in terms of line bundles on its 3 cover . To construct an Enriques surface from its 3 cover, we need a 3 surface together with a fixed-point-free involution so that the quotient ∕ becomes an Enriques surface. Thanks to the following theorem of Keum, we pick algebraic Kummer surfaces as candidates: Theorem 2]). An algebraic Kummer surface is a K3 cover of some Enriques surface.
When the covering map ∶ → ∕ = of an Enriques surface is fixed, we also need to ask which line bundles on are pullbacks of some line bundles on . The answer is also well-known, thanks to Horikawa. Next, we recall the construction of a Jacobian Kummer surface and intersection theory over it. Let be a generic curve of genus 2. Its Jacobian variety  = ( ) is an Abelian surface with Néron-Severi group Note that  has a natural involution with 16 fixed points. The complete linear system |2Θ| defines a morphism to ℙ 3 , which factors through the singular quartic ∕ (Kummer quartic) with 16 ordinary double points. The Kummer surface = () is defined as the minimal desingularization of ∕ . Throughout the rest of the paper, we fix the notations as follows. Notation 2.3. We follow the notation as in [1].

CONSTRUCTION USING KCOVERS
be a polarized Enriques surface, and let ∶ → = ∕ be its 3 cover. Suppose it verifies Conjecture 1.3, that is, has a line bundle which fits into Equation (1.3). Equation (1.3) can be completely translated into the numerical conditions on its 3 cover. Namely, we are interested in line bundles ∈ * Pic( ) ⊆ Pic( ) which verifies the equation where ∶= * . Note that if is ample and globally generated, then is also ample and globally generated, and vice versa. Now let be a Jacobian Kummer surface associated to a generic curve of genus 2. As mentioned in the previous section, some line bundles in Pic( ) require rational coefficients in 1 2 ℤ when we write it as linear combinations of and nodes . One typical example is called an even eight: Proof. Recall that Pic( ) is spanned by integral linear combinations of nodes and tropes , 6 . In particular, , ∈ 1 2 ℤ. We first check the condition * ≅ . A direct computation shows that * ≅ if and only if = 1 + 2 + 3 + 4 . We still need to show that ∈ Pic( ). Since 1 + 2 and 3 + 4 are divisible by 2 in Pic( ), but no other + are divisible by 2 [15,Proposition V.6]. Therefore, the coefficients are elements in 1 2 ℤ such that 1 + 2 ∈ ℤ and 3 + 4 ∈ ℤ. □ , find values ′ so that the line bundles and verify Equation (3.1).

By taking the substitutions
Equation Dividing both equations by 4 and taking their difference, we have  . Therefore, finding is equivalent to finding a solution ( , , , ) of this system of Diophantine equations (3.2), (3.3), where the corresponding satisfies the assumptions in Lemma 3.2.
In most cases, finding integral solutions of a system of Diophantine equations is extremely hard even though it has rationally parametrized solutions. Instead, we provide a sufficient condition on 's so that the system has a solution ( , , , ) which fits into all the conditions we need. Proposition 3.5. Let 1 , 2 , 3 , 4 ∈ 1 2 ℤ such that 1 + 2 ∈ ℤ, 3 + 4 ∈ ℤ, and ) , we obtain a number of polarized Enriques surfaces establishing the Borisov-Nuer conjecture as follows.
Together with a discussion on the moduli of (numerically) polarized Enriques surfaces, we get the following nonemptiness.
is a sum of two line bundles. Since the former one is very ample, and the later one is a multiple of a line bundle which induces an elliptic fibration over ℙ 1 (see [14,Fibration 7] and [10, Section 5.1]), their sum is indeed ample and globally generated. Moreover, the value is an integer, we conclude that there is a line bundle which verifies the equation by Proposition 3.5. For instance, we may take Proof. Note that any ( ′ , Since ′ is general, it is unnodal; it does not contain any smooth (−2)-curves. By [5, Proposition 2.1], ′ is an ′ -Ulrich line bundle as desired. □ Example 3.9. There are several possible choices of satisfying the assumptions of Proposition 3.5 and Theorem 3.7 when we fix the degree 2 . For instance, take 1 = 2 = 2 , 3 = 4 = 2 where , are positive integers. The line bundle is ample and globally generated with the self-intersection number 2 = 8 . Furthermore, the value is always an integer, so we are able to find a solution of Diophantine equations (3.2), (3.3). and + are integers, the left-hand side must be an even integer. Hence, there is no solution which satisfies the assumptions. In general, by a simple parity argument, one can easily check that the system (3.2), (3.3) does not have a solution ( , , , ) such that the corresponding satisfies the assumptions of Lemma 3.2 when the number 4 (which stands for 1 4 2 in the context) is not an even integer. This is the reason why it is not easy to verify the nonemptiness of Ξ when − 1 is not divisible by 4. For instance, we cannot verify that Borisov-Nuer conjecture holds for a Fano polarized Enriques surface ( , Δ) in the above arguments, since − 1 = Δ 2 = 10 is not divisible by 4. However, there might be plenty of chances to find a solution of Equation (3.1) using the same Jacobian Kummer surface. We only address a few more examples as evidence. We cannot guarantee that the following bundles are ample and/or globally generated, however, this aspect is not very important from the viewpoint of the original Borisov-Nuer conjecture.