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

TL;DR

This paper explores the Borisov-Nuer conjecture relating to the moduli spaces of polarized Enriques and K3 surfaces, proving non-emptiness of a specific intersection under certain divisibility conditions on the polarization.

Contribution

It demonstrates that the intersection of the locus  with the Borisov-Nuer divisor is non-empty when the genus-related parameter is divisible by 4, constructing explicit polarized Enriques surfaces.

Findings

The intersection   is non-empty for (g-1) divisible by 4.

Constructed polarized Enriques surfaces verify the Borisov-Nuer conjecture under divisibility conditions.

The conjecture holds for polarizations with square divisible by 4.

Abstract

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.

Equations

$$\begin{array}[]{cccl}\eta_{h}:\mathcal{M}_{En,h}^{a}&\to&\mathcal{F}_{g}\\ (Y,h=[H_{Y}])&\mapsto&(X_{Y},\sigma_{Y}^{*}H_{Y}).\\ \end{array}$$

$${\Sigma }_g:=\bigcup _{h^2=g-1}im({\eta }_h)\subseteq {\mathcal{F}}_g$$

$$H^i(X,\mathcal{E}(-j))=0\text{ for all }i\text{ and }1\le j\le n$$

$$(N-H_Y{)}^2=(N-2H_Y{)}^2=-2,$$

$$\left\{\begin{array}[]{rcl}H_{X}^{2}&=&2g-2,\\ M^{2}&=&4g-8,\\ H_{X}\cdot M{}&=&3g-3,\end{array}\right.$$

$$\Xi_{g}=\left\{\begin{array}[]{l|l}&\exists~{}\theta:X\to X\text{ fixed-point-free involution }\\ &\text{ such that }\theta^{*}H_{X}\simeq H_{X},\\ (X,H_{X})\in\mathcal{F}_{g}&\exists~{}M\in\operatorname{Pic}(X)\text{ such that }\theta^{*}M\simeq M,\\ &\text{ and }(X,H_{X},M)\in\mathcal{BN}_{g}.\end{array}\right\}$$

$$E_0$$

$$E_{ij}=E_{[p_i-p_j]}$$

$$\begin{array}[]{|ccc||ccc|}\hline\cr\text{Nodes}&&\text{Tropes}&\text{Nodes}&&\text{Tropes}\\ \hline\cr E_{0}&\leftrightarrow&T_{456}&E_{25}&\leftrightarrow&T_{246}\\ E_{12}&\leftrightarrow&T_{3}&E_{26}&\leftrightarrow&T_{136}\\ E_{13}&\leftrightarrow&T_{2}&E_{34}&\leftrightarrow&T_{356}\\ E_{14}&\leftrightarrow&T_{156}&E_{35}&\leftrightarrow&T_{346}\\ E_{15}&\leftrightarrow&T_{146}&E_{36}&\leftrightarrow&T_{126}\\ E_{16}&\leftrightarrow&T_{236}&E_{45}&\leftrightarrow&T_{6}\\ E_{23}&\leftrightarrow&T_{1}&E_{46}&\leftrightarrow&T_{5}\\ E_{24}&\leftrightarrow&T_{256}&E_{56}&\leftrightarrow&T_{4}\\ \hline\cr\end{array}$$

$$T_i$$

$$T_{ij6}$$

$$(M-H_X{)}^2=(M-2H_X{)}^2=-4$$

$$L-T_4-E_{14}-T_{146}-E_{46}$$

$${\theta }^{\ast }(E_{12}+E_{15}+E_{26}+E_{56})$$

$$\left\{\begin{array}[]{ccccccccc}F_{1}&=&E_{12}&+&E_{15}&+&E_{26}&+&E_{56}\\ F_{2}&=&E_{13}&+&E_{14}&+&E_{36}&+&E_{46}\\ F_{3}&=&E_{23}&+&E_{25}&+&E_{34}&+&E_{45}\\ F_{4}&=&E_{0}&+&E_{16}&+&E_{24}&+&E_{35}\\ \end{array}\right.$$

$${\theta }^{\ast }L$$

$${\theta }^{\ast }{F}_k$$

$$M=3L-F_1-F_2-F_4$$

$$\left\{\begin{array}[]{ccc}S&=&\beta_{1}^{\prime}-\beta_{1}\\ T&=&\beta_{2}^{\prime}-\beta_{2}\\ U&=&\beta_{3}^{\prime}-\beta_{3}\\ V&=&\beta_{4}^{\prime}-\beta_{4}\\ \end{array}\right.,$$

$$4(S+T+U+V{)}^2-8S^2-8T^2-8U^2-8V^2$$

$$4(\alpha -(S+T+U+V){)}^2-8({\beta }_1-S{)}^2-8({\beta }_2-T{)}^2$$

$$-8({\beta }_3-U{)}^2-8({\beta }_4-V{)}^2$$

$$(S+T+U+V{)}^2-2S^2-2T^2-2U^2-2V^2$$

$$2\alpha (S+T+U+V)-4{\beta }_{1}S-4{\beta }_{2}T-4{\beta }_{3}U-4{\beta }_{4}V-\frac{d}{4}$$

$$2S=\frac{1}{2({\beta }_3+{\beta }_4)}\left[({\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4{)}^2-2({\beta }_1^2+{\beta }_2^2+{\beta }_3^2+{\beta }_4^2)+2({\beta }_3-{\beta }_4)\right]\in \mathbb{Z}.$$

$$4({\beta }_3+{\beta }_4)S-({\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4{)}^2+2({\beta }_1^2+{\beta }_2^2+{\beta }_3^2+{\beta }_4^2)-2{\beta }_3+2{\beta }_4=0.$$

$$M:=({\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4+2S)L-({\beta }_1+S)F_1-({\beta }_2+S)F_2+\left({\beta }_3+\frac{1}{2}\right)F_3-\left({\beta }_4-\frac{1}{2}\right)F_4$$

$$\frac{1}{2({\beta }_3+{\beta }_4)}\left[({\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4{)}^2-2({\beta }_1^2+{\beta }_2^2+{\beta }_3^2+{\beta }_4^2)+2({\beta }_3-{\beta }_4)\right]=k$$

$$(M-H_X{)}^2=(M-2H_X{)}^2=-4$$

$$\frac{1}{2({\beta }_3+{\beta }_4)}\left[({\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4{)}^2-2({\beta }_1^2+{\beta }_2^2+{\beta }_3^2+{\beta }_4^2)+2({\beta }_3-{\beta }_4)\right]=m$$

$$-2S+2T+2U+2V=-1.$$

$$M=6L-3F_1-\frac{3}{2}(E_0+E_{13}+E_{14}+E_{16}+E_{25}+E_{34}+E_{36}+E_{46}).$$

$$L-T_1-T_{346}+E_{12}+E_{15}=\frac{1}{2}(E_0+E_{13}+E_{14}+E_{16}+E_{25}+E_{34}+E_{36}+E_{46}),$$

Full text

1. Introduction

Let $Y$ be an Enriques surface over $\mathbb{C}$, that is, a smooth projective surface with $p_{g}(Y)=q(Y)=0$ and $2K_{Y}=\mathcal{O}_{Y}$. The universal covering of $Y$ is given by an étale double cover map $\sigma_{Y}:X_{Y}\to Y$ where $X$ is a $K3$ surface. Hence, an Enriques surface $Y$ determines a pair $(X_{Y},\theta_{Y})$, where $X_{Y}$ is its $K3$ cover, and $\theta_{Y}$ is a fixed-point-free involution on $X_{Y}$ so that $\sigma_{Y}$ coincides with the quotient map $X_{Y}\to X_{Y}/\theta_{Y}$. In particular, studying Enriques surfaces $Y$ is equivalent to studying pairs $(X,\theta)$ of $K3$ surfaces $X$ and fixed-point-free involutions $\theta$ on $X$.

A polarized Enriques surface is a pair $(Y,H_{Y})$, where $Y$ is an Enriques surface and $L\in\operatorname{Pic}(Y)$ is an ample line bundle. A numerically polarized Enriques surface is a pair $(Y,[H_{Y}])$, where $[H_{Y}]\in\operatorname{Num}(Y)\cong U\oplus E_{8}(-1)$ denotes the numerical class of an ample line bundle $H_{Y}$ on $Y$. Fix a primitive vector $h\in U\oplus E_{8}(-1)$. Thanks to the lattice theory, Gritsenko and Hulek were able to give a construction of the moduli space ${\mathcal{M}}_{En,h}^{a}$ of numerically polarized Enriques surfaces with a polarization of type $h$ as an open subvariety of a modular variety $\mathcal{M}_{En,h}$, see [GH16] for details. It is a 10-dimensional quasi-projective variety, and the locus ${\mathcal{M}}_{En,h}^{nn}$ corresponding to unnodal surfaces (i.e., with no smooth $(-2)$-curves) is open. For an alternate approach to moduli spaces $\overline{\mathcal{E}}_{g,\phi}$ using the invariant $\phi$, we refer to [CDGK18].

Let us consider $\mathcal{F}_{g}$ the moduli space of polarized $K3$ surfaces of genus $g=h^{2}+1$. Note that $g$ is odd and $g\geq 5$. For any numerical type $h$, we have a natural map

[TABLE]

Then the locus

[TABLE]

consists of polarized $K3$ surfaces $(X,H_{X})$ which appear as pullbacks of polarized Enriques surfaces $(Y,H_{Y})$. Notice that, for any fixed degree $g-1$, there are only finitely many numerical types $h$ with $h^{2}=g-1$. Indeed, from [CDGK18, Proposition 3.4] it follows that the number of irreducible components of the moduli space $\overline{\mathcal{E}}_{g^{\prime},\phi}$ coincides with the number of possible simple decomposition types for $h$ for fixed values $h^{2}=2g^{\prime}-2$ and $\phi(h)=\phi$. Since $0<\phi^{2}\leq h^{2}$ by [CD89, Corollary 2.7.1], there are only finitely many possible choices of $\phi$, which implies the claim.

In this note, we discuss a conjecture of Borisov and Nuer on the Enriques lattice $\operatorname{Num}(Y)\simeq U\oplus E_{8}(-1)$, motivated by the Ulrich bundle existence problem, and connect it to the maps $\eta_{h}$. Let us briefly recall what are Ulrich bundles. Let $X\subset{\mathbb{P}}^{N}$ be a smooth projective variety of dimension $n$, and let $H={\mathcal{O}}_{X}(1)$ be a very ample line bundle on $X$. A vector bundle $\mathcal{E}$ on $X$ which satisfies the following cohomology vanishing condition

[TABLE]

is called an Ulrich bundle on $X$ [ESW03]. They have many interesting applications, in particular, they connect several different topics in algebra and geometry, see [ESW03, Bea18]. One important problem within this topic is to find an Ulrich bundle of smallest possible rank on a given variety. For an Enriques surface $Y$, together with a very ample line bundle $H_{Y}=\mathcal{O}_{Y}(1)$, it is known that $Y$ always carries an Ulrich bundle of rank $2$ [Bea16, Cas17]. On the other hand, Borisov and Nuer observed that the existence of an Ulrich line bundle $N$ on a polarized unnodal Enriques surface $(Y,H_{Y})$ is equivalent to the numerical condition

[TABLE]

that is, $H_{Y}$ 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 the equation (2). They conjectured that it is always possible to find such a line bundle $N$ for any choice of polarization $H_{Y}$, or even more, for any line bundle:

Conjecture 1** ([BN18, Conjecture 2.2]).**

For any line bundle $H$ on an Enriques surface $Y$, there is a line bundle $N\in\operatorname{Pic}(Y)$ such that $(N-H)^{2}=(N-2H)^{2}=-2$.

Suppose that $(Y,H_{Y})$ verifies the Borisov-Nuer conjecture; we have a line bundle $N$ on $Y$ which satisfies the above equation (2). We translate the conjecture in terms of line bundles on its $K3$ covers by observing the image under $\eta_{h}$ defined above. Let $\sigma:X\to Y=X/\theta$ be the universal cover, $H_{X}:=\sigma^{*}H_{Y}$, and let $M:=\sigma^{*}N$. The equation (2) is equivalent to

[TABLE]

where $g=H_{Y}^{2}+1\geq 5$ is an odd integer. Hence, if $(Y,H_{Y})$ and $N$ satisfy the equation (2), then the image $(X,H_{X})$ must lie in the Borisov-Nuer divisor $\mathcal{BN}_{g}$, which is a Noether-Lefschetz divisor $\mathcal{NL}_{2g-3,3g-3}\subset\mathcal{F}_{g}$ (the subscript stands for the numbers $\left(\frac{1}{2}M^{2}+1\right)$ and $H_{X}\cdot M$, respectively).

Note that a line bundle $M$ on the $K3$ cover $X$ is contained in $\sigma^{*}\operatorname{Pic}(Y)$ if and only if $\theta^{*}M\simeq M$ [Hor78]. In the case, the pushforward $\sigma_{*}M$ splits as a direct sum of two line bundles $N\oplus(N\otimes K_{Y})$, where $M\simeq\sigma^{*}N$. We consider the sublocus

[TABLE]

consisting of polarized $K3$ surfaces of genus $g$ which can be obtained by pullback of some polarized Enriques surface $(Y,H_{Y})$ together with a line bundle $N$ so that the triple $(Y,H_{Y},N)$ verifies the Conjecture 1. In particular, we have $\Xi_{g}\subseteq\Sigma_{g}\subset\mathcal{F}_{g}$. Since the Picard number $\rho(Y)$ of an Enriques surface $Y$ is $10$, both loci $\Xi_{g}$ and $\Sigma_{g}$ have high codimensions in $\mathcal{F}_{g}$. With this notation, Conjecture 1 implies:

Conjecture 2**.**

The two loci $\Xi_{g}$ and $\Sigma_{g}$ coincide.

Since the locus $\Xi_{g}$ is contained in the Borisov-Nuer divisor $\mathcal{BN}_{g}$ by definition, this conjecture admits the following much weaker version:

Question**.**

Is $\Sigma_{g}$ contained in $\mathcal{BN}_{g}$?

At the moment, the Borisov-Nuer conjecture is known for only a few examples: Fano polarization $\Delta$ and its multiple $k\Delta$ by Borisov and Nuer themselves [BN18, Theorem 2.4], and a degree $4$ polarization [AK17, Theorem 13]. In particular, $\Xi_{g}$ is nonempty when $g=5$ or $g=11$. To have a better understanding, it is worthwhile to observe $\Xi_{g}$, and to collect more evidences for the Borisov-Nuer conjecture.

In this paper, we construct examples of points in $\Xi_{g}$ for various values of $g$. Suppose that Conjecture 1 holds for a numerically polarized Enriques surface $(Y,[H_{Y}])$ with $H_{Y}$ of type $h$. Since all the Enriques surfaces $Y$ have the same lattice structure $\operatorname{Num}(Y)\simeq U\oplus E_{8}(-1)$, we immediately have that Conjecture 1 holds for every numerically polarized Enriques surface $(Y^{\prime},[H_{Y}^{\prime}])\in{\mathcal{M}}_{En,h}^{a}$. Hence, it suffices to construct only one numerically polarized Enriques surface $(Y,h)$ from the moduli space $\mathcal{M}_{En,h}^{a}$ which makes Conjecture 1 hold. The key ingredient is a Jacobian Kummer surface $X=Km(C)$ of a general curve $C$ of genus $2$, similar as in [AK17]. Such a Jacobian Kummer surface has plenty of technical merits, for instance:

• •

$X$ has a fixed-point-free involution $\theta$, that is, $X$ is the $K3$ cover of some Enriques surface $Y$;

• •

intersection theory of $X$ is well-understood;

• •

the pullback homomorphism $\theta^{*}:\operatorname{Pic}(X)\to\operatorname{Pic}(X)$ is well-understood;

• •

the Picard number $\rho(X)$ is quite big, so there are more chances to find a certain line bundle.

The main result of this paper is the nonemptyness of the locus $\Xi_{g}$ for various values $g$ as follows, see Theorem 11:

Theorem**.**

When $g-1$ is divisible by $4$, the locus $\Xi_{g}$ is nonempty. In other words, for any given $k>0$ and any Enriques surface $Y$, there is an ample and globally generated line bundle $H_{Y}$ and a line bundle $N$ on $Y$ such that $H_{Y}^{2}=4k$ and $(N-H_{Y})^{2}=(N-2H_{Y})^{2}=-2$.

The outline of the paper is the following. In Section 2, we review some basic facts on Enriques surfaces, Jacobian Kummer surfaces as $K3$ 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 $(g-1)$ is not divisible by $4$.

2. 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 the equation (2) on an Enriques surface $Y$ in terms of line bundles on its $K3$ cover $X$. To construct an Enriques surface from its $K3$ cover, we need a $K3$ surface $X$ together with a fixed-point-free involution $\theta$ so that the quotient $X/\theta$ becomes an Enriques surface. Thanks to the following theorem of Keum, we pick algebraic Kummer surfaces as candidates:

Lemma 3** ([Keu90, Theorem 2]).**

An algebraic Kummer surface is a K3 cover of an Enriques surface.

When the covering map $\sigma:X\to X/\theta=Y$ of an Enriques surface is fixed, we also need to ask which line bundles on $X$ are pullbacks of some line bundles on $Y$. The answer is also well-known, thanks to Horikawa.

Lemma 4** ([Hor78, Theorem 5.1]).**

Let $X$ be a $K3$ surface, $\theta:X\to X$ be a fixed-point-free involution, and $\sigma:X\to Y=X/\theta$ be the $2:1$ étale cover. Then the image of the map $\sigma^{*}:\operatorname{Pic}(Y)\to\operatorname{Pic}(X)$ is the set of line bundles $M$ in $X$ such that $\theta^{*}M\simeq M$.

Next, we recall the construction of a Jacobian Kummer surface and intersection theory over it. Let $C$ be a generic curve of genus $2$. Its Jacobian variety $\mathcal{A}=J(C)$ is an Abelian surface with Néron-Severi group $NS(\mathcal{A})=\mathbb{Z}\cdot[\Theta]$ with $\Theta^{2}=2$. Note that $\mathcal{A}$ has a natural involution $\iota$ with $16$ fixed point. The complete linear system $|2\Theta|$ defines a morphism to $\mathbb{P}^{3}$, which factors through the singular quartic $\mathcal{A}/\iota$ (Kummer quartic) with $16$ ordinary double points. The Kummer surface $X=Km(\mathcal{A})$ is defined as the minimal desingularization of $\mathcal{A}/\iota$. Throughout the rest of the paper, we fix the notations as follows.

Notation 5**.**

We follow the notation as in [AK17].

• •

$C$ : a generic curve of genus 2 with 6 Weierstrass points $p_{1},\cdots,p_{6}\in C$;

• •

$X=Km(C)$ : Jacobian Kummer surface associated to $C$, which is the minimal desingularization of $J(C)/\iota$;

• •

$\theta:X\to X$ : a fixed-point-free involution so called “switch” induced by the even theta characteristic $[p_{4}+p_{5}-p_{6}]$;

• •

$\sigma:X\to Y=X/\theta$ : the quotient map so that $Y$ is an Enriques surface;

• •

$L$ : the line bundle induced by the hyperplane section of the singular quartic $J(C)/\iota\subseteq\mathbb{P}^{3}$;

• •

$E_{0},E_{ij}~{}(1\leq i<j\leq 6)$ : sixteen $(-2)$-curves called nodes;

• •

$T_{i}~{}(1\leq i\leq 6),T_{ij6}~{}(1\leq i<j\leq 5)$ : sixteen $(-2)$-curves called tropes.

Note that $L^{2}=4,L\cdot E_{0}=L\cdot E_{ij}=0$, and two distinct nodes do not intersect.

Let us describe the nodes and the tropes more precisely. Following the notation in [Oha09], the 16 nodes are labeled by the corresponding 2–torsion points in the Jacobian $\mathcal{A}=J(C)$:

[TABLE]

The tropes are labeled using their associated theta–characteristics of $C$ [Oha09], e.g. $T_{i}=T_{[p_{i}]}$ corresponds to $[p_{i}]$ and $T_{ijk}=T_{[p_{i}+p_{j}-p_{k}]}$ corresponds to $[p_{i}+p_{j}-p_{k}]$ for any $i<j<k$. Note that $T_{ijk}=T_{\ell mn}$ if $\{i,j,k\}\cup\{\ell,m,n\}=\{1,\ldots,6\}$.

Also note that the pullback $\theta^{*}$ swaps the nodes $E_{\alpha}$ and the tropes $T_{\alpha+\beta}$ in the following way, cf. [Muk12] and [Oha09, Section 4, Section 5]:

[TABLE]

where the corresponding tropes are

[TABLE]

for $1\leq i\leq 6$ and

[TABLE]

for $1\leq i<j\leq 5$, where $\{l,m,n\}$ is the complement of $\{i,j\}$ in $\{1,2,3,4,5\}$ (see [Oha09, Lemma 4.1]).

It is well-known that $\{E_{0},E_{ij},T_{i},T_{ij6}\}$ spans $\operatorname{Pic}(X)$ [Keu97, Lemma 3.1], and hence $\{L,E_{0},E_{ij}\}$ spans $\operatorname{Pic}(X)\otimes\frac{1}{2}\mathbb{Z}$ if we allow $\frac{1}{2}\mathbb{Z}$ coefficients. For simplicity, we mostly consider a linear combination of $L,E_{0},E_{ij}$ in $\frac{1}{2}\mathbb{Z}$ coefficients, however, we have to carefully choose the coefficients so that the linear combination gives an element in $\operatorname{Pic}(X)$.

3. Construction using K3 covers

Let $(Y,H_{Y})$ be a polarized Enriques surface, and let $\sigma:X\to Y=X/\theta$ be its $K3$ cover. Suppose it verifies Conjecture 2, that is, $Y$ has a line bundle $N$ which fits into the equation (2). The equation (2) can be completely translated into the numerical conditions on its $K3$ cover. Namely, we are interested in line bundles $M\in\sigma^{*}\operatorname{Pic}(Y)\subseteq\operatorname{Pic}(X)$ which verifies the equation

[TABLE]

where $H_{X}:=\sigma^{*}H_{Y}$. Note that if $H_{Y}$ is ample and globally generated, then $H_{X}$ is also ample and globally generated, and vice versa.

Now let $X$ be a Jacobian Kummer surface associated to a generic curve $C$ of genus $2$. As mentioned in the previous section, some line bundles in $\operatorname{Pic}(X)$ require rational coefficients in $\frac{1}{2}\mathbb{Z}$ when we write it as linear combinations of $L$ and nodes $E_{ij}$. One typical example is called an even eight:

Lemma 6**.**

The set of $8$ nodes $\{E_{0},E_{16},E_{23},E_{24},E_{25},E_{34},E_{35},E_{45}\}$ forms an even eight, that is, $\left(E_{0}+E_{16}+E_{23}+E_{24}+E_{25}+E_{34}+E_{35}+E_{45}\right)$ is divisible by $2$ in $\operatorname{Pic}(X)$.

Proof.

It is straightforward from a direct computation

[TABLE]

∎

Also note that the complementary set of nodes $\{E_{12},E_{13},E_{14},E_{15},E_{26},E_{36},E_{46},E_{56}\}$ also forms an even eight. Since

[TABLE]

and by similar computations, grouping them by those 4 line bundles makes the problem easier. Let $F_{\bullet}$ be the sum of four nodes $E_{ij}$, namely,

[TABLE]

We have

[TABLE]

Consider a linear combination of the form $M=\alpha L-\beta_{1}F_{1}-\beta_{2}F_{2}-\beta_{3}F_{3}-\beta_{4}F_{4}$ as a special case. First, we need to check when $M$ becomes a $\theta^{*}$-invariant line bundle on $X$.

Lemma 7**.**

A linear combination $M=\alpha L-\beta_{1}F_{1}-\beta_{2}F_{2}-\beta_{3}F_{3}-\beta_{4}F_{4}$ is a line bundle in $\operatorname{Pic}(X)$ such that $\theta^{*}M\simeq M$ if and only if $\beta_{i}\in\frac{1}{2}\mathbb{Z}$, $\beta_{1}+\beta_{2}\in\mathbb{Z}$, $\beta_{3}+\beta_{4}\in\mathbb{Z}$, and $\alpha=\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}$.

Proof.

Recall that $\operatorname{Pic}(X)$ is spanned by integral linear combinations of nodes $E_{ij}$ and tropes $T_{i},T_{ij6}$. In particular, $\alpha,\beta_{i}\in\frac{1}{2}\mathbb{Z}$. We first check the condition $\theta^{*}M\simeq M$. A direct computation shows that $\theta^{*}M\simeq M$ if and only if $\alpha=\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}$.

We still need to show that $M\in\operatorname{Pic}(X)$. Since $F_{1}+F_{2}$ and $F_{3}+F_{4}$ are divisible by 2 in $\operatorname{Pic}(X)$, but no other $F_{i}+F_{j}$ are divisible by $2$ [Meh06, Proposition V.6] , hence the coefficients $\beta_{i}$ are elements in $\frac{1}{2}\mathbb{Z}$ such that $\beta_{1}+\beta_{2}\in\mathbb{Z}$ and $\beta_{3}+\beta_{4}\in\mathbb{Z}$. ∎

Example 8**.**

Let $\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=\frac{1}{2}$, and $\alpha=\sum\beta_{i}=2$. The line bundle $H_{X}=2L-\frac{1}{2}(F_{1}+F_{2}+F_{3}+F_{4})$ satisfies the assumptions in Lemma 7, and defines an embedding of $X$ into $\mathbb{P}^{5}$ as the intersection of $3$ quadrics [Shi77, Theorem 2.5]. Such a Kummer surface $(X,H_{X})$ carries a line bundle $M$ such that $\theta^{*}M\simeq M$, namely,

[TABLE]

as in [AK17, proof of Theorem 13]. Furthermore, $H_{X}$ and $M$ satisfies the equation (3) as desired.

Let $H_{X}=\alpha L-\sum_{k=1}^{4}\beta_{k}F_{k}$, and let $M=\alpha^{\prime}L-\sum_{k=1}^{4}\beta_{k}^{\prime}F_{k}$. Suppose that both $H_{X}$ and $M$ satisfies the assumptions in Lemma 7. Now our question becomes:

Question**.**

For a given ample polarization $H_{X}=\alpha L-\sum_{k=1}^{4}\beta_{k}F_{k}$, find values $\beta_{i}^{\prime}$ so that the line bundles $H_{X}$ and $M$ verify the equation (3).

By taking the substitutions

[TABLE]

the equation (3) gives the system of two quadratic Diophantine equations, namely:

[TABLE]

Dividing both equations by 4 and taking their difference, we have

[TABLE]

where $\alpha=\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}$ and $d=H_{X}^{2}=4\alpha^{2}-8(\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2})$. Therefore, finding $M$ is equivalent to finding a solution $(S,T,U,V)$ of this system of Diophantine equations (4), (5), where the corresponding $M$ satisfies the assumptions in Lemma 7.

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 $\beta_{i}$’s so that the system has a solution $(S,T,U,V)$ which fits into all the conditions we need.

Proposition 9**.**

Let $\beta_{1},\beta_{2},\beta_{3},\beta_{4}\in\frac{1}{2}\mathbb{Z}$ such that $\beta_{1}+\beta_{2}\in\mathbb{Z}$, $\beta_{3}+\beta_{4}\in\mathbb{Z}$, and

[TABLE]

Then the above system of Diophantine equations has a solution $(S,T,U,V)=(S,S,\frac{1}{2},-\frac{1}{2})$ so that $\beta_{1}^{\prime},\cdots,\beta_{4}^{\prime}$ satisfy the assumptions in Lemma 7.

Proof.

It is clear that $(S,S,\frac{1}{2},-\frac{1}{2})$ is a solution for the equation (4). Substitute into the equation (5), we have a univariable linear equation

[TABLE]

It is straightforward that such a solution $(S,T,U,V)=(S,S,\frac{1}{2},-\frac{1}{2})$ provides $\beta_{1}^{\prime},\beta_{2}^{\prime},\beta_{3}^{\prime},\beta_{4}^{\prime}$ which satisfies the assumptions in Lemma 7. ∎

By taking suitable quadruples $(\beta_{1},\beta_{2},\beta_{3},\beta_{4})$, we obtain a number of polarized Enriques surfaces establishing the Borisov-Nuer conjecture as follows.

Proposition 10**.**

Suppose that $H_{X}=(\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4})L-\beta_{1}F_{1}-\beta_{2}F_{2}-\beta_{3}F_{3}-\beta_{4}F_{4}$ is an ample and globally generated line bundle on $X$ such that $\beta_{1},\cdots,\beta_{4}$ satisfy the assumptions in Proposition 9. Then there is a polarized Enriques surface $(Y,H_{Y})$ and a line bundle $N$ on $Y$ such that $H_{X}^{2}=2H_{Y}^{2}$ and $(N-H_{Y})^{2}=(N-2H_{Y})^{2}=-2$. In particular, the Borisov-Nuer conjecture holds for $(Y,H_{Y})$.

Proof.

Let $S=\frac{1}{4(\beta_{3}+\beta_{4})}\left[(\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4})^{2}-2(\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2})+2(\beta_{3}-\beta_{4})\right]$. Proposition 9 implies that $(S,S,\frac{1}{2},-\frac{1}{2})$ is a solution of the system of Diophantine equations (4), (5). Hence, the line bundle

[TABLE]

verifies the conditions $\theta^{*}M\simeq M$ and $(M-H_{X})^{2}=(2H_{X}-M)^{2}=-4$.

By Lemma 4, there are line bundles $H_{Y}$ and $N$ on an Enriques surface $Y=X/\theta$ such that $\sigma^{*}H_{Y}=H_{X}$, $\sigma^{*}N=M$ where $\sigma:X\to Y=X/\theta$ is the quotient map. Since $\sigma_{*}H_{X}=H_{Y}\oplus(H_{Y}\otimes K_{Y})$ and $\sigma_{*}M=N\oplus(N\otimes K_{Y})$, we conclude that $(N-H_{Y})^{2}=(2H_{Y}-N)^{2}=-2$. ∎

Together with a discussion on the moduli of (numerically) polarized Enriques surfaces, we get the following non-emptiness.

Theorem 11**.**

The locus $\Xi_{g}$ contained in the Borisov-Nuer divisor $\mathcal{BN}_{g}\subset\mathcal{F}_{g}$ of polarized $K3$ surfaces of degree $H_{X}^{2}=2g-2$ is nonempty when $2g-2$ is divisible by $8$. In particular, there is a numerically polarized Enriques surface $(Y,h=[H_{Y}])\in\mathcal{M}_{En,h}^{a}$ which verifies the Borisov-Nuer conjecture when $h^{2}=g-1$ is divisible by $4$. Moreover, the conjecture also holds for every $(Y^{\prime},[H_{Y^{\prime}}])\in\mathcal{M}_{En,h}^{a}$.

Proof.

Let $X$ be a general Jacobian Kummer surface as above. It suffices to construct a pair $(H_{X},M)$ of line bundles on $X$ determined by the values $\beta_{i}$’s and $\beta_{i}^{\prime}$’s satisfying Proposition 9. Suppose $g=4k+1$ so that $H_{X}^{2}=8k$ is divisible by 8. We pick $H_{X}=(k+1)L-\frac{k}{2}(F_{1}+F_{2})-\frac{1}{2}(F_{3}+F_{4})$ so that $H_{X}^{2}=8k,\beta_{1}=\beta_{2}=\frac{k}{2},\beta_{3}=\beta_{4}=\frac{1}{2}$. Note that $H_{X}=\left[2L-\frac{1}{2}(F_{1}+F_{2}+F_{3}+F_{4})\right]+(k-1)\left[L-\frac{1}{2}(F_{1}+F_{2})\right]$ 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 ${\mathbb{P}}^{1}$ (see [Kum14, Fibration 7] and [GS16, Section 5.1]), their sum $H_{X}$ is indeed ample and globally generated.

Moreover, the value

[TABLE]

is an integer, we conclude that there is a line bundle $M$ which verifies the equation

[TABLE]

by Proposition 9. For instance, we may take $M=(2k+1)L-k(F_{1}+F_{2})-F_{3}$. ∎

Corollary 12**.**

Let $(Y,h=[H_{Y}])\in\mathcal{M}_{En,h}^{a}$ be a numerically polarized Enriques surface appearing in Theorem 11. Let $(Y^{\prime},[H_{Y}^{\prime}])\in\mathcal{M}_{En,h}^{a}$ be a generic element. Then the Enriques surface $Y^{\prime}$ has an $H_{Y}^{\prime}$-Ulrich line bundle, in the sense of [AK17, Definition 1].

Proof.

Note that any $(Y^{\prime},[H_{Y}^{\prime}])\in\mathcal{M}_{En,h}^{a}$ carries a line bundle $N^{\prime}\in\operatorname{Pic}(Y^{\prime})$ such that $(N^{\prime}-H_{Y}^{\prime})^{2}=(N^{\prime}-2H_{Y}^{\prime})^{2}=-2$. Since $Y^{\prime}$ is general, it is unnodal; it does not contain any smooth $(-2)$-curves. By [BN18, Proposition 2.1], $N^{\prime}$ is an $H_{Y}^{\prime}$-Ulrich line bundle as desired. ∎

Example 13**.**

There are several possible choices of $H_{X}$ satisfying the assumptions of Proposition 9 and Theorem 11 when we fix the degree $H_{X}^{2}$. For instance, take $\beta_{1}=\beta_{2}=\frac{m}{2}$, $\beta_{3}=\beta_{4}=\frac{n}{2}$ where $m,n$ are positive integers. The line bundle $H_{X}:=(m+n)L-\frac{m}{2}(F_{1}+F_{2})-\frac{n}{2}(F_{3}+F_{4})$ is ample and globally generated with the self-intersection number $H_{X}^{2}=8mn$. Furthermore, the value

[TABLE]

is always an integer, so we are able to find a solution of Diophantine equations (4), (5).

Remark 14**.**

The system of Diophantine equations (4), (5) needs not to have a desired solution. For example, let $\beta_{1}=1$, $\beta_{2}=\beta_{3}=\beta_{4}=0$. Then the second equation (5) becomes

[TABLE]

Since $-S+T=(\beta_{2}^{\prime}-\beta_{1}^{\prime})-(\beta_{2}-\beta_{1})$ and $U+V$ 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 (4), (5) does not have a solution $(S,T,U,V)$ such that the corresponding $M$ satisfies the assumptions of Lemma 7 when the number $\frac{d}{4}$ (which stands for $\frac{1}{4}H_{X}^{2}$ in the context) is not an even integer. This is the reason why it is not easy to verify the nonemptiness of $\Xi_{g}$ when $g-1$ is not divisible by $4$. For instance, we cannot verify that Borisov-Nuer conjecture holds for a Fano polarized Enriques surface $(Y,\Delta)$ in the above arguments, since $g-1=\Delta^{2}=10$ is not divisible by $4$.

However, there might be plenty of chances to find a solution of the equation (3) using the same Jacobian Kummer surface. We only address a few more examples as evidence. We cannot guarantee that the following bundles $H_{X}$ are ample and/or globally generated, however, this aspect is not very important from the viewpoint of the original Borisov-Nuer conjecture.

1. (i)

Let $H_{X}=4L-2F_{1}-F_{2}-\frac{1}{2}F_{3}-\frac{1}{2}F_{4}$ so that $\theta^{*}H_{X}\simeq H_{X}$ and $H_{X}^{2}=20$. We take $M$ as

[TABLE]

Since

[TABLE]

$M$ is a line bundle on $X$. Furthermore, $M$ satisfies $\theta^{*}M\simeq M$ and $(M-H_{X})^{2}=(M-2H_{X})^{2}=-4$. Hence, there is an Enriques surface $Y$ and two line bundles $H_{Y},N$ with $H_{Y}^{2}=10$ such that $(N-H_{Y})^{2}=(2H_{Y}-N)^{2}=-2$. 2. (ii)

Let $H_{X}=6L-3F_{1}-2F_{2}-\frac{1}{2}F_{3}-\frac{1}{2}F_{4}$ so that $\theta^{*}H_{X}\simeq H_{X}$ and $H_{X}^{2}=36$. We take $M$ as

[TABLE]

We have $M\in\operatorname{Pic}(X)$, $\theta^{*}M\simeq M$, and $H_{X},M$ satisfy the equation (3). 3. (iii)

Let $H_{X}=8L-4F_{1}-3F_{2}-\frac{1}{2}F_{3}-\frac{1}{2}F_{4}$. We have $\theta^{*}H_{X}\simeq H_{X}$ and $H_{X}^{2}=52$. We take $M$ as

[TABLE]

We have $M\in\operatorname{Pic}(X)$, $\theta^{*}M\simeq M$, and $H_{X},M$ satisfy the equation (3).

Acknowledgments**.**

MA was partly supported by a grant of Ministery of Research and Innovation, CNCS–UEFISCDI, project number PN-III-P4-ID-PCE-2016-0030, within PNCDI III. YK thanks Simion Stoilow Institute of Mathematics of the Romanian Academy (IMAR) for the hospitality during his visit. He was partly supported by Project I.6 of SFB-TRR 195 “Symbolic Tools in Mathematics and their Application” of the German Research Foundation (DFG).