Ulrich line bundles on Enriques surfaces with a polarization of degree four

Aprodu, Marian; Kim, Yeongrak

doi:10.1007/s11565-017-0269-z

Cited by 9 publications

(10 citation statements)

References 20 publications

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“…Throughout the rest of the paper, we fix the notations as follows. Notation We follow the notation as in [1]. C : a generic curve of genus 2 with 6 Weierstrass points

p_{1}, \dots, p_{6} \in C

;

X = K m (C)

: Jacobian Kummer surface associated to C , which is the minimal desingularization of

J (C) / ι

;

θ : X \to X

: a fixed‐point‐free involution, so‐called “switch”, induced by the even theta characteristic

[p_{4} + p_{5} - p_{6}]

;

σ : X \to Y = X / θ

: the quotient map so that Y is an Enriques surface; L : the line bundle induced by the hyperplane section of the singular quartic

J false(C false) / ι \subseteq {double-struckP}^{3}

;

E_{0}, E_{i j} false(1 \leq i < j \leq 6 false)

: sixteen ( − 2)‐curves called nodes ;

T_{i} false(1 \leq i \leq 6 false), T_{i j 6} false(1 \leq i < j \leq 5 false)

: sixteen ( − 2)‐curves called tropes , which satisfy the following relations

\begin{matrix} right T_{i} & = & left \frac{1}{2} (L - E_{0} - \sum_{k \neq i} E_{i k}) & left for 1 \leq i \leq 6, and \\ right T_{i j 6 ...} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…The line bundle

H_{X} = 2 L - \frac{1}{2} (F_{1} + F_{2} + F_{3} + F_{4})

satisfies the assumptions in Lemma 3.2, and defines an embedding of X into

P^{5}

as the intersection of 3 quadrics [18, Theorem 2.5]. Such a Kummer surface

(X, H_{X})

carries a line bundle M such that

θ^{*} M ≅ M

, namely,

M = 3 L - F_{1} - F_{2} - F_{4}

as in [1, proof of Theorem 13]. Furthermore,

H_{X}

and M satisfies Equation (3.1) as desired.…”

Section: Construction Using K3 Coversmentioning

confidence: 99%

“…Let

(Y^{'}, true[H_{Y}^{'} true]) \in {scriptM}_{E n, h}^{a}

be any numerically polarized unnodal Enriques surface. Then the Enriques surface

Y^{'}

has an

H_{Y}^{'}

‐Ulrich line bundle, in the sense of [1, Definition 1].…”

Section: Construction Using K3 Coversmentioning

confidence: 99%

“…At the moment, the Borisov–Nuer conjecture is known for only a few examples: Fano polarization Δ and its multiple

k Δ

by Borisov and Nuer themselves [5, Theorem 2.4], and a degree 4 polarization [1, Theorem 13]. In particular,

Ξ_{g}

is nonempty when

g = 5

g = 11

.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, it suffices to construct only one numerically polarized Enriques surface

(Y, true[H_{Y} true])

from the moduli space

M_{E n, h}^{a}

which makes Conjecture 1.1 hold. The key ingredient is a Jacobian Kummer surface

X = K m (C)

of a general curve C of genus 2, similar as in [1]. Such a Jacobian Kummer surface has plenty of technical merits, for instance: X has a fixed‐point‐free involution θ, that is, X is the K 3 cover of some Enriques surface Y ; intersection theory of X is well‐understood; the pullback homomorphism

θ^{*} : Pic false(X false) \to Pic false(X false)

is well‐understood; the Picard number

ρ (X)

is quite big, so there are more chances to find a certain line bundle. …”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Aprodu

Kim

2020

Mathematische Nachrichten

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We discuss the Borisov-Nuer conjecture in connection with the canonical maps from the moduli spaces  ,ℎ of polarized Enriques surfaces with fixed ℎ ∈ ⊕ 8 (−1) to the moduli space  of polarized 3 surfaces of genus with = ℎ 2 + 1, and we exhibit a naturally defined locus Σ ⊂  . One direct consequence of the Borisov-Nuer conjecture is that Σ would be contained in a particular Noether-Lefschetz divisor in  , which we call the Borisov-Nuer divisor and we denote by  . In this short note, we prove that Σ ∩  is non-empty whenever ( − 1) is divisible by 4. To this end, we construct polarized Enriques surfaces ( , ) , with 2 divisible by 4, which verify the conjecture. In particular, when we consider the moduli space of (numerically) polarized Enriques surfaces which contains such ( , ), the conjecture also holds for any other polarized Enriques surface ( ′ , ′ ) contained in the same moduli. K E Y W O R D SBorisov-Nuer conjecture, Enriques surface, Jacobian Kummer surface, numerically polarized Enriques surface M S C ( 2 0 1 0 ) 14D20, 14J28

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“…Throughout the rest of the paper, we fix the notations as follows. Notation We follow the notation as in [1]. C : a generic curve of genus 2 with 6 Weierstrass points