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

TL;DR

This paper establishes a precise correspondence between Markov fractions and the slopes of exceptional vector bundles on the projective plane, simplifying previous proofs and clarifying their numerical relationships.

Contribution

It demonstrates that Markov fractions are exactly the slopes of exceptional bundles on b6, providing a simpler proof of Rudakov's result linking ranks of these bundles to Markov numbers.

Findings

Markov fractions correspond to slopes of exceptional bundles

Ranks of exceptional bundles are Markov numbers

Simplified proof of Rudakov's result

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 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 $\mathbb P^2$ are Markov numbers.