Mutations of fake weighted projective planes

TL;DR

This paper characterizes one-step mutations of weighted projective planes via T-singularities and Diophantine equations, extending previous work on lattice polytope mutations and their associated toric deformations.

Contribution

It provides a complete classification of mutations between weighted projective planes in terms of singularities and Diophantine equations, expanding the understanding of toric variety deformations.

Findings

Complete characterization of mutations via T-singularities

Identification of Diophantine equations governing weights

Extension of previous mutation results to weighted projective planes

Abstract

In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives 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 characterisation of such mutations in terms of T-singularities. We show also that the weights involved satisfy Diophantine equations, generalising results of Hacking-Prokhorov.