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 $\mathbb P<sup>2$ studied in 1980s by Dr`ezet 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 $\mathbb P<sup>2$ are Markov numbers.