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Mutations of Fake Weighted Projective Planes

Part of: Surfaces and higher-dimensional varieties Polytopes and polyhedra Diophantine equations

Published online by Cambridge University Press:  10 June 2015

Mohammad E. Akhtar  and
Alexander M. Kasprzyk
Mohammad E. Akhtar
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK, (mohammad.akhtar03@imperial.ac.uk; a.m.kasprzyk@imperial.ac.uk)
Alexander M. Kasprzyk
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK, (mohammad.akhtar03@imperial.ac.uk; a.m.kasprzyk@imperial.ac.uk)
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Abstract

In previous work by Coates, Galkin and the authors, the notion of mutation between lattice polytopes was introduced. Such mutations give rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterization of such mutations in terms of T-singularities. We also show that the weights involved satisfy Diophantine equations, generalizing results of Hacking and Prokhorov.

Keywords

weighted projective spacemutationT-singularity

MSC classification

Primary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Secondary: 14J45: Fano varieties 11D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 271 - 285
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. Akhtar, M. Coates, T., Galkin, S. and Kasprzyk, A. M., Minkowski polynomials and mutations, SIGMA: Symmetry Integrability Geom. Methods Applic. 8 (2012), 094.CrossRefGoogle Scholar
2. Buczyńska, W., Fake weighted projective spaces, preprint (arxiv.org/abs/0805.1211, 2008).Google Scholar
3. Conrads, H., Weighted projective spaces and reflexive simplices, Manuscr. Math. 107(2) (2002), 215–227.CrossRefGoogle Scholar
4. Emerson, E. I., Recurrent sequences in the equation DQ 2 = R 2 +N , Fibonacci Quarterly 7(3) (1969), 231–242.Google Scholar
5. Hacking, P. and Prokhorov, Y., Smoothable del Pezzo surfaces with quotient singularities, Compositio Math. 146(1) (2010), 169–192.Google Scholar
6. Ilten, N. O., Mutations of Laurent polynomials and flat families with toric fibers, SIGMA: Symmetry Integrability Geom. Methods Applic. 8 (2012), 047.Google Scholar
7. Kasprzyk, A. M., Bounds on fake weighted projective space, Kōdai Math. J. 32 (2009), 197–208.Google Scholar
8. Kasprzyk, A. M. and Nill, B., Fano polytopes, in Strings, gauge fields, and the geometry behind: the legacy of Maximilian Kreuzer (ed. Rebhan, A. Katzarkov, L., Knapp, J., Rashkov, R. and Scheidegger, E.), pp. 349–364 (World Scientific, 2012).CrossRefGoogle Scholar
9. Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. Math. 91(2) (1988), 299–338.Google Scholar
10. Markoff, A., Sur les formes quadratiques binaires indéfinies, Math. Annalen 17(3) (1880), 379–399.CrossRefGoogle Scholar
11. Rosenberger, G., Uber die diophantische Gleichung ax 2 + by 2 + cz 2 = dxyz , J. Reine Angew. Math. 305 (1979), 122–125.Google Scholar