In the context of K3 mirror symmetry, the Greene–Plesser orbifolding method constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in P3, starting from a special one-parameter family ...of K3 varieties known as the quartic Dwork pencil. We show that certain K3 double covers obtained from the three-parameter family of quartic Kummer surfaces associated with a principally polarized abelian surface generalize the relation of the Dwork pencil and the quartic mirror family. Moreover, for the three-parameter family we compute a formula for the rational point-count of its generic member and derive its transformation behavior with respect to (2,2)-isogenies of the underlying abelian surface.
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This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, Washington. ...Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics. The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic K3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
Nikulin Involutions and the CHL String Clingher, Adrian; Malmendier, Andreas
Communications in mathematical physics,
09/2019, Volume:
370, Issue:
3
Journal Article
Peer reviewed
Open access
We study certain even-eight curve configurations on a specific class of Jacobian elliptic K3 surfaces with lattice polarizations of rank ten. These configurations are associated with K3 double ...covers, some of which are elliptic but not Jacobian elliptic. Several non-generic cases corresponding to K3 surfaces of higher Picard rank are also discussed. Finally, the results and the construction in question are interpreted in the context of the string dualities linked with the eight-dimensional CHL string.
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Abstract
We study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank 14, 2‐elementary lattices. Three such lattices exist, namely, , , and . As part of our study, we ...provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.
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This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, ...Washington.Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics.The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic $K$3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
We construct non-geometric compactifications using the F-theory dual of the heterotic string compactified on a two-torus, together with a close connection between Siegel modular forms of genus two ...and the equations of certain K3 surfaces. The modular group mixes together the Kähler, complex structure, and Wilson line moduli of the torus yielding weakly coupled heterotic string compactifications which have no large radius interpretation.
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Let ... be any point in the moduli space of genus-two curves ... and ... its field of moduli. We provide a universal equation of a genus-two curve ... defined over ..., corresponding to ..., where ...... and ... satisfy a quadratic ... such that ... and ... are given in terms of ratios of Siegel modular forms. The curve ... is defined over the field of moduli ... if and only if the quadratic has a ...-rational point .... We discover some interesting symmetries of the Weierstrass equation of .... This extends previous work of Mestre and others. ProQuest: ... denotes formulae omitted.
We show that the duality between F-theory and the CHL string in seven dimensions defines algebraic correspondences between K3 surfaces polarized by the rank-ten lattices
H
⊕
N
and
H
⊕
E
8
(
-
2
)
. ...In the special case when the F-theory admits an additional anti-symplectic involution or, equivalently, the CHL string admits a symplectic one, both moduli spaces coincide. In this case, we derive an explicit parametrization for the F-theory compactifications dual to the CHL string, using an auxiliary genus-one curve, based on a construction given by André Weil.
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We classify all Jacobian elliptic fibrations on K3 surfaces with finite automorphism group. We also classify all Jacobian elliptic fibrations with finite Mordell–Weil group on K3 surfaces with ...infinite automorphism group and 2-elementary Néron–Severi lattice. As part of the classification, we compute the lattice theoretic multiplicities of all Jacobian elliptic fibrations in both cases.
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