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

A. P. Veselov

TL;DR

This work identifies Markov fractions as precisely the slopes of exceptional vector bundles on $\mathbb P^2$, linking the old geometric classification by Drèzet–Le Potier and Rudakov to Springborn's rational tree. The key idea is to match the Drèzet–Le Potier function $\epsilon(x)$ with the Springborn mediant rule, thereby proving that the set of exceptional slopes coincides with the Markov fractions and giving a streamlined proof that the ranks of exceptional bundles are Markov numbers. The analysis highlights the $Aff_1(\mathbb Z)$-invariance of slope data, extends the slope-functions to real numbers with Minkowski-type dynamics, and connects to Diophantine and hyperbolic-geometry phenomena such as McShane identities and Lagrange numbers. Overall, the paper forges a bridge between algebraic geometry of vector bundles on $\mathbb P^2$ and arithmetic/diophantine structures emergent from Markov theory, with implications for related surfaces and conjectures in the broader area of stability conditions and derived categories.

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.

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

TL;DR

This work identifies Markov fractions as precisely the slopes of exceptional vector bundles on , linking the old geometric classification by Drèzet–Le Potier and Rudakov to Springborn's rational tree. The key idea is to match the Drèzet–Le Potier function with the Springborn mediant rule, thereby proving that the set of exceptional slopes coincides with the Markov fractions and giving a streamlined proof that the ranks of exceptional bundles are Markov numbers. The analysis highlights the -invariance of slope data, extends the slope-functions to real numbers with Minkowski-type dynamics, and connects to Diophantine and hyperbolic-geometry phenomena such as McShane identities and Lagrange numbers. Overall, the paper forges a bridge between algebraic geometry of vector bundles on and arithmetic/diophantine structures emergent from Markov theory, with implications for related surfaces and conjectures in the broader area of stability conditions and derived categories.

Abstract

We show that the Markov fractions introduced recently by Boris Springborn are precisely the slopes of the exceptional vector bundles on 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 are Markov numbers.
Paper Structure (6 sections, 8 theorems, 64 equations, 4 figures)

This paper contains 6 sections, 8 theorems, 64 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Markov fractions
  3. Exceptional slopes and Markov fractions
  4. Extension to real numbers and Minkowski question mark function
  5. Concluding remarks
  6. Acknowledgements

Key Result

Proposition 2.1

The numbers $p,q$, which are defined recursively on the Conway topograph by the Springborn local rule (mfrac) with the initial data $\frac{0}{1}$ and $\frac{1}{2},$ are coprime integers. The denominators $q$ of Markov fractions are the corresponding Markov numbers.

Figures (4)

  • Figure 1: The Farey tree of fractions between 0 and 1.
  • Figure 2: The Markov fraction tree with the Springborn local rule.
  • Figure 3: Markov numbers on the Conway topograph.
  • Figure 4:

Theorems & Definitions (13)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 3 more