Cubic surfaces and Borcherds products

Allcock, Daniel; Freitag, Eberhard

doi:10.1007/s00014-002-8340-4

Cited by 40 publications

(93 citation statements)

References 10 publications

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“…In terms of hermitian symmetric domains this corresponds to the embedding B ֒→ D of the 10-dimensional ball in a 20-dimensional type IV domain D (induced by the natural embedding of groups U (1, 10) ⊂ O(2, 20)). We construct φ by restricting to B an automorphic form φ on D (for a similar construction see [6]). …”

Section: Consider a Minimal Blow-up Of B/γ Such That The Pull-back Ofmentioning

confidence: 99%

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

Casalaina-Martin¹,

Laza²

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

118

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Abstract. A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.

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Section: Consider a Minimal Blow-up Of B/γ Such That The Pull-back Ofmentioning

confidence: 99%

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

Casalaina-Martin¹,

Laza²

2009

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

118

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“…Thus, the geometrical structure of K 3 perfectly fits the structure of W (E 6 ) into its non-solvable maximal subgroups. See also [44,45].…”

Section: Discussionmentioning

confidence: 99%

Unitary Reflection Groups for Quantum Fault Tolerance

Planat¹,

Kibler²

2010

Jnl of Comp & Theo Nano

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Abstract.This paper explores the representation of quantum computing in terms of unitary reflections (unitary transformations that leave invariant a hyperplane of a vector space). The symmetries of qubit systems are found to be supported by Euclidean real reflections (i.e., Coxeter groups) or by specific imprimitive reflection groups, introduced (but not named) in a recent paper [Planat M and Jorrand Ph 2008, J Phys A: Math Theor 41, 182001]. The automorphisms of multiple qubit systems are found to relate to some Clifford operations once the corresponding group of reflections is identified. For a short list, one may point out the Coxeter systems of type B 3 and G 2 (for single qubits), D 5 and A 4 (for two qubits), E 7 and E 6 (for three qubits), the complex reflection groups G(2 l , 2, 5) and groups No 9 and 31 in the Shephard-Todd list. The relevant fault tolerant subsets of the Clifford groups (the Bell groups) are generated by the Hadamard gate, the π/4 phase gate and an entangling (braid) gate [Kauffman L H and Lomonaco S J 2004 New J. of Phys. 6, 134]. Links to the topological view of quantum computing, the lattice approach and the geometry of smooth cubic surfaces are discussed.

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“…We take this triple to be (h 12 , h 34 , h 56 ). Then we may take α = h and we see that these are roots of the D 4 -system spanned by h 12 The following notion is the root system analogue of its namesake introduced by Allcock and Freitag [1].…”

Section: Proposition 41 (Macdonald) Let R Be a Root System H The Cmentioning

confidence: 99%

“…While keeping that in mind we nevertheless wish to mention the influential book by Manin [18], the Astérisque volume by Dolgachev-Ortland [10], Naruki's construction of a smooth compactification of the moduli space of marked cubic surfaces [16], the determination of its Chow groups in [5], the Lecture Note by Hunt [15] and Yoshida's revisit of the Coble covariants [19]. The ball quotient description of the moduli space of cubic surfaces by Allcock, Carlson and Toledo [2], combined with Borcherds' theory of modular forms, led Allcock and Freitag [1] to construct an embedding of the moduli space of marked cubic surfaces, which coincides with the map given by the Coble invariants [13], [14]. We now briefly review the organization of the paper.…”

Section: Introductionmentioning

confidence: 99%

Del Pezzo Moduli via Root Systems

Colombo¹,

Geemen²,

Looijenga³

2009

Progress in Mathematics

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ABSTRACT. Coble defined in his 1929 treatise invariants for cubic surfaces and quartic curves. We reinterpret these in terms of the root systems of type E 6 and E 7 that are naturally associated to these varieties, thereby giving some of his results a more intrinsic treatment. Our discussion is uniform for all Del Pezzo surfaces of degree 2,3,4 and 5.To Professor Yuri Manin, for his 70th birthday.

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Cubic surfaces and Borcherds products

Cited by 40 publications

References 10 publications

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

The moduli space of cubic threefolds via degenerations of the intermediate Jacobian

Unitary Reflection Groups for Quantum Fault Tolerance

Del Pezzo Moduli via Root Systems

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