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.

41 pages, 13 figures, embedded data file