Classifications of Elliptic Fibrations of a Singular K3 Surface

Bertin, Marie José; Garbagnati, Alice; Hortsch, Ruthi; Lecacheux, Odile; Mase, Makiko; Salgado, Cecília; Whitcher, Ursula

doi:10.1007/978-3-319-17987-2_2

Cited by 12 publications

(21 citation statements)

References 23 publications

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“…Moreover, if π is branched over two different smooth fibers, τ (C) = C implies that τ is an involution of C, and thus D is a section of the elliptic fibration E R . Hence if D is a component of a fiber one must have τ (C) = C, i.e., case (2) with m = 0.…”

Section: Rational Curves On K3 Surfacesmentioning

confidence: 99%

“…Since X is a double cover of an extremal rational elliptic surface R branched on smooth fibers, the reducible fibers of the inherited fibration E X contribute with 16 components to a set of generators of NS(X). If X is generic among such surfaces then it lies in a 2-dimensional family and hence NS(X) has 2 We exclude rational elliptic surfaces with (2I * 0 ). These have a 1-dimensional moduli space.…”

Section: Double Covers Of Extremal Rational Elliptic Surfacesmentioning

confidence: 99%

“…It is therefore a natural problem to classify such fibrations. This has been done in the past three decades, via different methods by several authors, see for instance [15], [14], [7], [2], [3], [6] and [1]. Recently, the second and third authors have proposed a new method to classify elliptic fibrations on K3 surfaces which arise as double cover of rational elliptic surfaces.…”

Section: Introductionmentioning

confidence: 99%

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Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

Cantoral-Farfán¹,

Garbagnati²,

Salgado³

et al. 2020

Preprint

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We study K3 surfaces over a number field k which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.

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Section: Rational Curves On K3 Surfacesmentioning

confidence: 99%

Section: Double Covers Of Extremal Rational Elliptic Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

Cantoral-Farfán¹,

Garbagnati²,

Salgado³

et al. 2020

Preprint

Self Cite

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“…12 Elliptic fibrations with a global section, and their Weierstrass forms, of an attractive K3 surface whose transcendental lattice has the intersection matrix 4 0 0 2 are classified in [58]. Elliptic fibrations with a global section of an attractive K3 surface whose transcendental lattice has the intersection matrix 6 0 0 2 are studied in [59].…”

Section: Enhancement To Models With U (1)mentioning

confidence: 99%

Enhancements in F-theory models on moduli spaces of K3 surfaces with ADE rank 17

Kimura

Mizoguchi

2018

Progress of Theoretical and Experimental Physics

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We study the moduli of elliptic K3 surfaces with a section with the ADE rank 17. While the Picard number of a generic K3 surface in such moduli space is 19, the Picard number is enhanced to 20 at special points in the moduli. K3 surfaces become attractive K3 surfaces at these points. Either of the following two situations occurs at such special points: i) the Mordell-Weil rank of an elliptic K3 surface is enhanced, or ii) the gauge symmetry is enhanced. The first case i) is related to the appearance of a U (1) gauge symmetry. In this note, we construct the moduli of K3 surfaces with ADE types E 7 D 10 and A 17 . We determine some of the special points at which K3 surfaces become attractive in the moduli of K3 surfaces with ADE types E 7 D 10 and A 17 . We investigate the gauge symmetries in F-theory compactifications on attractive K3 surfaces which correspond to such special points in the moduli times a K3 surface. U (1) gauge symmetry arises in some F-theory compactifications. arXiv:1712.08539v3 [hep-th]

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“…In Section 4, we recall Borcherds method, and apply it to 11 singular K3 surfaces whose transcendental lattices are listed in Table 4.1. Recently, many geometric studies of singular K3 surfaces of small discriminant have appeared (see, for example, [3], [12], [21], [40]). We summarize the computational data for these 11 singular K3 surfaces in Table 4.2.…”

Section: Introductionmentioning

confidence: 99%

Enriques involutions on singular K3 surfaces of small discriminants

Shimada¹,

Veniani²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We classify Enriques involutions on a K3 surface, up to conjugation in the automorphism group, in terms of lattice theory. We enumerate such involutions on singular K3 surfaces with transcendental lattice of discriminant smaller than or equal to 36. For 11 of these K3 surfaces, we apply Borcherds method to compute the automorphism group of the Enriques surfaces covered by them. In particular, we investigate the structure of the two most algebraic Enriques surfaces.ICHIRO SHIMADA AND DAVIDE CESARE VENIANI but X 7 does. Hence, following Vinberg, we call the Enriques surfaces covered by X 7 the most algebraic Enriques surfaces.Theorem 1.0.1. The singular K3 surface X 7 of discriminant 7 has exactly two Enriques involutionsε I andε II up to conjugation in Aut(X 7 ). Let Y I and Y II be the quotient Enriques surfaces corresponding toε I andε II , respectively. Then Aut(Y I ) is finite of order 8, and Aut(Y II ) is finite of order 24.Nikulin [26] and Kondo [19] classified all complex Enriques surfaces whose automorphism group is finite. It turns out that these Enriques surfaces are divided into 7 classes I, II, . . . , VII, which we call Nikulin-Kondo type. See Kondo [19] for the properties of these Enriques surfaces. Corollary 1.0.2. The most algebraic Enriques surfaces have finite automorphism groups and their Nikulin-Kondo types are I and II.In Section 6 of this paper, we give explicit models of the most algebraic Enriques surfaces Y I and Y II as Enriques sextic surfaces.Remark 1.0.3. The Néron-Severi lattice and the automorphism group of X 7 were determined by Ujikawa [39]. Elliptic fibrations on X 7 were studied by Harrache-Lecacheux [12] and Lecacheux [21].Remark 1.0.4. Mukai [23] also realized that X 7 has Enriques involutions that produce Enriques surfaces of Nikulin-Kondo type I and II.Ohashi [27] gave a lattice theoretic method to enumerate Enriques involutions on certain K3 surfaces. He then classified in [28] all Enriques involutions on the Kummer surface Km(Jac(C)) associated with the jacobian variety of a generic curve C of genus 2. We refine and generalize Ohashi's method. Our main result, namely Theorem 3.1.9, applies to any K3 surface, and we use it in the case of singular K3 surfaces to compile Table 3.1.For some K3 surfaces X, the group Aut(X) can be calculated by Borcherds method ([4], [5]); for instance, Kondo [20] implemented it in order to compute Aut(Km(Jac(C))). We apply Borcherds method in order to calculate the automorphism group of some of singular K3 surfaces X, and to write the action of Aut(X) on the nef chamber of X explicitly. Building on this data, we enumerate all Enriques involutions up to conjugation, and, using also a result of the preprint [7] (see Section 2.9), we calculate the automorphism group of the Enriques surfaces covered by these K3 surfaces.Note that the enumeration of Enriques involutions by Ohashi's method and by Borcherds method are carried out independently. The results are, of course, consistent. We hope that these methods will be applied to many other K3 surface...

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Classifications of Elliptic Fibrations of a Singular K3 Surface

Abstract: We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface X whose transcendental lattice is isometric to 6 ⊕ 2 .

Cited by 12 publications

References 23 publications

Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

Enhancements in F-theory models on moduli spaces of K3 surfaces with ADE rank 17

Enriques involutions on singular K3 surfaces of small discriminants

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