Unfocused notes on the Markoff equation and T-Singularities

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.