Markov fractions and the slopes of the exceptional bundles on ${\mathbb{P}}^2$

Abstract

We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on ${\mathbb{P}}^2$ studied in 1980s by Drèzet and Le Potier and by Rudakov. In particular, we provide a simpler proof of Rudakov's result claiming that the ranks of the exceptional bundles on ${\mathbb{P}}^2$ are Markov numbers.

Slightly revised and extended version