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On the geometry of C3/∆27 and del Pezzo surfaces

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Abstract

We clarify some aspects of the geometry of a resolution of the orbifold \( X = {{{\mathbb{C}^3}} \mathord{\left/{\vphantom {{{\mathbb{C}^3}} {{\Delta_{27}}}}} \right.} {{\Delta_{27}}}} \), the noncompact complex manifold underlying the brane quiver standard model recently proposed by Verlinde and Wijnholt. We explicitly realize a map between X and the total space of the canonical bundle over a degree 1 quasi del Pezzo surface, thus defining a desingularization of X. Our analysis relys essentially on the relationship existing between the normalizer group of ∆27 and the Hessian group and on the study of the behaviour of the Hesse pencil of plane cubic curves under the quotient.

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Authors and Affiliations

  1. Dipartimento di Fisica, Università degli Studi dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Sergio L. Cacciatori

  2. Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133, Milano, Italy

    Marco Compagnoni

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  1. Sergio L. Cacciatori
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  2. Marco Compagnoni
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Correspondence to Marco Compagnoni.

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ArXiv ePrint: 1001.2893

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Cacciatori, S.L., Compagnoni, M. On the geometry of C3/∆27 and del Pezzo surfaces. J. High Energ. Phys. 2010, 78 (2010). https://doi.org/10.1007/JHEP05(2010)078

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