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Mutations of Fake Weighted Projective Planes

Published online by Cambridge University Press:  10 June 2015

Mohammad E. Akhtar
Department of Mathematics, Imperial College London, London SW7 2AZ, UK, (;
Alexander M. Kasprzyk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK, (;


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.

Research Article
Copyright © Edinburgh Mathematical Society 2016 

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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 (, 2008).Google Scholar
3. Conrads, H., Weighted projective spaces and reflexive simplices, Manuscr. Math. 107(2) (2002), 215227.CrossRefGoogle Scholar
4. Emerson, E. I., Recurrent sequences in the equation DQ 2 = R 2 +N , Fibonacci Quarterly 7(3) (1969), 231242.Google Scholar
5. Hacking, P. and Prokhorov, Y., Smoothable del Pezzo surfaces with quotient singularities, Compositio Math. 146(1) (2010), 169192.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), 197208.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. 349364 (World Scientific, 2012).CrossRefGoogle Scholar
9. Kollár, J. and Shepherd-Barron, N. I., Threefolds and deformations of surface singularities, Invent. Math. 91(2) (1988), 299338.Google Scholar
10. Markoff, A., Sur les formes quadratiques binaires indéfinies, Math. Annalen 17(3) (1880), 379399.CrossRefGoogle Scholar
11. Rosenberger, G., Uber die diophantische Gleichung ax 2 + by 2 + cz 2 = dxyz , J. Reine Angew. Math. 305 (1979), 122125.Google Scholar