Elliptic Fibrations on Covers of the Elliptic Modular Surface of Level 5

Abstract

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, R 5,5 . Such surfaces have a natural elliptic fibration induced by the fibration on R 5,5 . Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on R 5,5 . This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type I 5 and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.

Calabi–Yau 4-folds of Borcea–Voisin type from F-theory

“…In case m = 4, there are two different choices for π : S 2 → P 1 . One of them is characterized by the presence of a 5-torsion section for π : S 2 → P 1 and in this case the K3 surface S 2 is a 2 : 1 cover of the rational surface with a level 5 structure, see [BDGMSV17]. We observe that if π : S 2 → P 1 admits a 5-torsion section, the same is true for E.…”

On two-elementary K3 surfaces with finite automorphism group

“…In [8,45] it was proved that a complex algebraic K3 surface with Néron-Severi lattice H ⊕ E 8 (−1) ⊕ E 8 (−1) admits a birational model isomorphic to an Inose quartic. Other examples of equations relating the elliptic fibrations of K3 surfaces with 2-elementary Néron-Severi lattices and double sextics or quartic hypersurfaces were provided in [1,20]. In [25] the authors considered the moduli spaces of K3 surfaces that are obtained from the 4-cyclic covers of P 2 branched along quartic curves.…”

“…Here, L is a choice of even indefinite lattice of signature (1, ρ L −1), with 1 ≤ ρ L ≤ 20. Two L-polarized K3 surfaces (X , i) and (X ′ , i ′ ) are said to be isomorphic 1 , if there exists an isomorphism α∶ X → X ′ and a lattice isometry β ∈ O(L), such that α * ○ i ′ = i ○ β, where α * is the appropriate morphism at cohomology level.…”

Fields of Definition of Elliptic Fibrations on Covers of Certain Extremal Rational Elliptic Surfaces

“…A reason to explore elliptic fibrations on X i , i = 2, 3, 4, 9 is that they have different behavior with respect to the cover involution of X i → R i . Fibrations that are preserved by this involution are easier to describe via linear systems of curves on a rational surface, and one can also exhibit a Weierstrass equation for those as pointed out in [1] and [6]. In particular, on X 3 and X 4 , which can be identified, we have two different involutions (induced by the covers X 4 → R 4 and X 3 → R 3 ) and the behavior of each fibration on X 3 X 4 with respect to these two involutions can be different.…”

“…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.…”