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The Root Lattice \(A_{2}\) in the Construction of Substitution Tilings and Singular Hypersurfaces

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Applications of Computer Algebra (ACA 2015)

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Abstract

The analysis of the critical points of a one-parameter family of polynomials allows us to define sets of pseudolines in the fundamental region of the affine Weyl group associated with the root lattice \(A_{2}\). The pseudolines are transformed into configurations of lines containing the prototiles of substitution tilings with n-fold symmetry. The configurations of lines have been used recently to obtain hypersurfaces with many singularities. Calabi–Yau threefolds can be constructed from resolutions of some of the singular hypersurfaces.

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Acknowledgements

This work was partially supported by Consejería de Educación del Principado de Asturias, Spain (UO-15-INVES-38).

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Authors and Affiliations

  1. Facultad de Ciencias, Universidad de Oviedo, 33007, Oviedo, Spain

    Juan García Escudero

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  1. Juan García Escudero
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Correspondence to Juan García Escudero .

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Editors and Affiliations

  1. Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  2. Institute of Mathematics, University of Valladolid, Valladolid, Spain

    Edgar Martínez-Moro

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Escudero, J.G. (2017). The Root Lattice \(A_{2}\) in the Construction of Substitution Tilings and Singular Hypersurfaces. In: Kotsireas, I., Martínez-Moro, E. (eds) Applications of Computer Algebra. ACA 2015. Springer Proceedings in Mathematics & Statistics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-319-56932-1_7

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