Unfocused notes on the Markoff equation and T-Singularities

TL;DR

This paper classifies minimal resolutions of certain weighted projective plane singularities related to the Markoff equation, exploring their structure through continued fractions and the effects of mutations, revealing a Cantor set as a limit.

Contribution

It provides a complete classification of resolutions for these singularities using continued fractions and analyzes their behavior under mutations, introducing new geometric insights.

Findings

Resolutions are classified via continued fractions similar to Frobenius's work.

Mutations affect the structure of resolutions and lead to a Cantor set of limits.

A novel connection between Markoff equation solutions and geometric resolutions is established.

Abstract

We consider minimal resolutions of the singularities for weighted projective planes of type $\mathbb{P}(e^2, f^2, g^2)$, where $e, f, g$ satisfy the Markoff equation $ e^2 + f^2 + g^2 = 3efg$. We give a complete classification of such resolutions in terms of continued fractions similar to classical work of Frobenius. In particular, we investigate the behaviour of resolutions under mutations and describe a Cantor set emerging as limits of continued fractions.