On the Cynk-Hulek criterion for crepant resolutions of double covers

Generalization of the Symmetric Starter of the Orthogonal Double Covers of the Complete Bipartite Graph

Menoufia Journal of Electronic Engineering Research ◽

10.21608/mjeer.2006.64856 ◽

2006 ◽

Vol 16 (2) ◽

pp. 171-177

Author(s):

S. A. El-Serafi ◽

R. A. El-Shanawany ◽

M. Sh. Higazy

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Double Covers

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Double cover K3 surfaces of Hirzebruch surfaces

Advances in Geometry ◽

10.1515/advgeom-2020-0034 ◽

2021 ◽

Vol 21 (2) ◽

pp. 221-225

Author(s):

Taro Hayashi

Keyword(s):

Rational Surface ◽

K3 Surfaces ◽

Double Cover ◽

Double Covering ◽

Smooth Surfaces ◽

Branch Locus ◽

Hirzebruch Surfaces ◽

Double Covers

Abstract General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.

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Folding of Hitchin Systems and Crepant Resolutions

International Mathematics Research Notices ◽

10.1093/imrn/rnaa375 ◽

2021 ◽

Author(s):

Florian Beck ◽

Ron Donagi ◽

Katrin Wendland

Keyword(s):

Fixed Point ◽

Lie Algebras ◽

Integrable Systems ◽

Trace Back ◽

Dynkin Diagrams ◽

Or Groups ◽

Hitchin Systems ◽

Singular Fibers ◽

Crepant Resolutions ◽

Intermediate Jacobian

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.

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