A note on deformations and mutations of fake weighted projective planes

TL;DR

This paper explores the deformation theory of certain algebraic surfaces, connecting Markov equation solutions, Fano triangles, and Minkowski summands through polarized T-varieties, providing explicit deformation descriptions.

Contribution

It introduces a new approach using polarized T-varieties and divisorial polytopes to explicitly describe deformations of weighted projective planes.

Findings

Explicit deformation descriptions via Minkowski summands.

Connection between Markov solutions and Fano triangle mutations.

Generalization of previous deformation results.

Abstract

It has been shown by Hacking and Prokhorov that if the projective surface X with quotient singularities and self-intersection number 9 has a smoothing to the projective plane, then X is the general fiber of a Q-Gorenstein deformation of the weighted projective plane with weights giving solutions to the Markov equation. This result has been understood and generalized by combinatorial mutations of Fano triangles by Akhtar, Coates, Galkin, and Kasprzyk. In this note, we study this result by utilizing polarized T-varieties and describe the associated deformation explicitly in terms of certain Minkowski summands of so-called divisorial polytopes.