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MINIMALITY AND MUTATION-EQUIVALENCE OF POLYGONS

Published online by Cambridge University Press:  15 August 2017

ALEXANDER KASPRZYK
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK; a.m.kasprzyk@nottingham.ac.uk
BENJAMIN NILL
Affiliation:
Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Postschließfach 4120, 39016 Magdeburg, Germany; benjamin.nill@ovgu.de
THOMAS PRINCE
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK; t.prince12@imperial.ac.uk

Abstract

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We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine representatives for all mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface; or (ii) a surface with only singularities of type $1/3(1,1)$.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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