Lattices and K3 surfaces of degree 6

Sterk, Hjm Hans

doi:10.1016/0024-3795(95)00150-p

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…Similar computation is known for the case M =< 6 > (see [39]). We have the following types of elliptic fibrations on surfaces from K M 3 : Ẽ8 , Ẽ8 ; D16 ; Ẽ8 , Ẽ7 , Ã2 ; D14 , Ã2 , Ã1 ; D10 , D6 ; D8 , Ẽ7 , Ã1 ; Ã15 , Ẽ6 , Ẽ6 , Ã5 ; Ã11 , D5 , Ã1 ; Ã9 , D7 .…”

Section: Let Us Now Compute the Global Monodromy Group γsupporting

confidence: 65%

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Mirror symmetry for lattice polarizedK3 surfaces

Dolgachev¹

1996

J Math Sci

330

488

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Introduction. There has been a recent explosion in the number of mathematical publications due to the discovery of a certain duality between some families of Calabi-Yau threefolds made by a group of theoretical physicists (see [11,26] for references). Roughly speaking this duality, called mirror symmetry, pairs two families F and F * of Calabi-Yau threefolds in such a way that the following properties are satisfied:MS1 The choice of the mirror family F * involves the choice of a boundary point ∞ of a compactification F of the moduli space for F at which the monodromy is "maximally unipotent". MS2 For each V ∈ F and V ′ ∈ F * the Hodge numbers satisfy * Research supported in part by a NSF grant.

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Section: Let Us Now Compute the Global Monodromy Group γsupporting

confidence: 65%

“…Similar computation is known for the case M =< 6 > (see [39]). We have the following types of elliptic fibrations on surfaces from K M 3 :…”

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confidence: 65%

Mirror symmetry for lattice polarizedK3 surfaces

Dolgachev¹

1996

J Math Sci

330

488

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“…When the arrangement is reflective in the above sense, then the closure of D in Z bb supports an effective Cartier divisor if and only if we can assign in a Γ-equivariant manner to each chamber σ a nonzero vector v σ ∈ σ ∩L(Q) which does not lie in any member of H. Example 5.12. A beautiful (and now classical) example to which these results apply is due to Conway (Chapter 27 of [9]) and Borcherds [5] respectively: take for L(Z) any even unimodular lattice of signature (1,25) (there is only one isomorphism class of these), let H be the collection of all hyperplanes perpendicular to the (−2)-vectors in L(Z), C ⊂ L(R) one of the two cones and let Γ be the subgroup of the orthogonal group of L(Z) which preserves C. Since the direct sum of the Leech lattice (with a minus sign) and a hyperbolic lattice is even unimodular of signature (1,25), it follows that there exists a primitive isotropic vector v ∈ L(Z) in the closure of the positive cone such that v ⊥ /Zv is isomorphic to (minus) the Leech lattice. It also follows that such vectors make up a Γ-orbit.…”

Section: It Is Easily Checked That If σ ′′ Is a Third Member Of σ(H)mentioning

confidence: 99%

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

Looijenga¹

2003

Duke Math. J.

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Abstract. We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the proj of an algebra of meromorphic automorphic forms. When that complement has a moduli space interpretation, then what we get is often a compactification obtained by means of geometric invariant theory. We illustrate this with several examples: moduli spaces of polarized K3 and Enriques surfaces and the semi-universal deformation of a triangle singularity.We also discuss the question when a type IV arrangement is definable by an automorphic form.

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“…Indeed taking the primitive embedding of E 6 ⊕ A 1 into the Niemeier lattice E 3 8 defined by the primitive embeddings E 6 ֒→ E 8 and A 1 ֒→ E 8 , we get the above elliptic fibration since (E 6 ) ⊥ E8 = A 2 and (A 1 ) ⊥ E8 = E 7 [8]. It corresponds to the second fibration among the ten jacobian elliptic fibrations obtained by Sterk [18].…”

Section: Thus We Can Compute the Norm Ofmentioning

confidence: 89%

Transcendental lattices of certain singular K3 surfaces

Bertin¹,

Lecacheux²

2021

Preprint

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We compute the transcendental lattices of the singular K3 surfaces belonging to three pencils of K3 surfaces, namely the Apéry-Fermi pencil with transcendental lattice U ⊕ 12 , the Verrill's pencil with transcendental lattice U ⊕ 6 and another pencil linked to Verrill's pencil with transcendental lattice U ⊕ 24 . Many corollaries are deduced. For example, some singular K3 surfaces belong to different pencils or are Kummer surfaces of K3 surfaces of another pencil.

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Lattices and K3 surfaces of degree 6

Cited by 7 publications

References 10 publications

Mirror symmetry for lattice polarizedK3 surfaces

Mirror symmetry for lattice polarizedK3 surfaces

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

Transcendental lattices of certain singular K3 surfaces

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