On the Classification of Elliptic Fibrations modulo Isomorphism on K3 Surfaces with large Picard Number

Abstract

Motivated by a problem originating in string theory, we study elliptic fibrations on K3 surfaces with large Picard number modulo isomorphism. We give methods to determine upper bounds for the number of inequivalent K3 surfaces sharing the same frame lattice. For any given Neron--Severi lattice $S_X$, such a bound on the `multiplicity' can be derived by investigating the quotient of the isometry group of $S_X$ by the automorphism group. The resulting bounds are strongest for large Picard numbers and multiplicities of unity do indeed occur for a number of K3 surfaces with Picard number 20. Under a few extra conditions, a more refined analysis is also possible by explicitly studying the embedding of $S_X$ into the even unimodular lattice ${\mathrm{II}}_{1,25}$ and exploiting the detailed structure of the isometry groups of $S_X$ and ${\mathrm{II}}_{1,25}$. We illustrate these methods in examples and derive bounds for the number of elliptic fibrations on Kummer surfaces of Picard numbers 17 and 20. As an intermediate step, we also discuss coarser classification schemes and review known results.

77 pages, 8 figures, 9 tables, includes a review on classifications of elliptic fibrations on K3 surfaces

Cited by in corpus (11)

• F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches
• Gauged 2-form Symmetries in 6D SCFTs Coupled to Gravity
• Multiple Fibrations in Calabi-Yau Geometry and String Dualities
• The Noether-Lefschetz Problem and Gauge-Group-Resolved Landscapes: F-Theory on K3 x K3 as a Test Case
• Heterotic-Type IIA Duality and Degenerations of K3 Surfaces
• Counting elliptic fibrations on K3 surfaces
• W=0 Complex Structure Moduli Stabilization on CM-type K3 x K3 Orbifolds:---Arithmetic, Geometry and Particle Physics---
• Classifications of elliptic fibrations of a singular K3 surface
• New perspectives in the duality of M-theory, heterotic strings, and F-theory
• Some observations about isogenies between $K3$ surfaces
• Jacobian fibrations on the singular K3 surface of discriminant 3