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

A.P. Veselov

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

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Pith's one-line read The slopes of the exceptional vector bundles on the projective plane are exactly the Markov fractions.

desk verdict A clean identification of Springborn's Markov fractions with the exceptional slopes on P2; the core algebra checks out and the main external input is the DLP classification. read the letter →

arxiv 2501.06779 v2 pith:QKFUEYZY submitted 2025-01-12 math.NT math.AG

A.P. Veselov This is my paper · ORCID

classification math.NTmath.AG MSC 14J6011J06

keywords MarkovfractionsexceptionalvectorbundlesslopesnumbersSpringbornmediantDrézet–LePotierclassificationConwaytopographDiophantineapproximation

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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Springborn's Markov fractions and the slopes of the exceptional vector bundles on the complex projective plane are the same set of rational numbers. The paper proves this by showing that the Drézet–Le Potier recursion, which describes all exceptional slopes as values of a function on dyadic rationals, is exactly the Springborn mediant rule that builds the Markov fraction tree. If correct, the result unifies two independent-looking classifications: one from algebraic geometry (rigid stable bundles on $\mathbb{P}^2$) and one from Diophantine approximation (rationals with denominators given by Markov numbers). It also gives a simpler proof of Rudakov's theorem that the ranks of these bundles are Markov numbers, and it ties the Frobenius unicity conjecture to a statement about exceptional bundles being determined by their rank.

What carries the argument

The load-bearing object is the Springborn mediant, which replaces the Farey mediant $(p_1/q_1)\oplus(p_2/q_2)=(p_1+p_2)/(q_1+q_2)$ with $(p_1q_1+p_2q_2)/(q_1^2+q_2^2)$; iterated from $0/1$ and $1/2$ on the Farey tree, it generates the Markov fractions. The other half of the machinery is the Drézet–Le Potier recursion (14), which defines $\epsilon$ on dyadic rationals. The proof's key identity (15) shows that, for neighbouring Markov fractions, the Drézet–Le Potier rule and the Springborn rule produce the same rational; the algebra uses the Markov equation $q_1^2+q_2^2+q_3^2=3q_1q_2q_3$ and the tree relation (9) from Lemma 2.2. Once the two recursions are identified, the Drézet–Le Potier classification transfers the equality to the full slope set.

What would settle it

Generate the Markov fraction tree to a fixed depth and compute the Drézet–Le Potier function $\epsilon$ on all dyadic rationals with denominator up to a matching bound; the theorem predicts the two lists coincide, so the first rational appearing in only one list would refute it. Alternatively, exhibiting an exceptional bundle on $\mathbb{P}^2$ whose slope is not a Markov fraction, or a Markov fraction for which no exceptional bundle exists, would settle the matter directly.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1: the set $\mathcal{E}$ of slopes of exceptional vector bundles on $\mathbb{P}^2$ coincides with the set of all Markov fractions. Exceptional bundles are the stable bundles with $\operatorname{Ext}^1(E,E)=0$; their slope is the ratio of first Chern class to rank, $\mu(E)=c_1(E)/r(E)$, and it is known from [8] that the slope determines the bundle. Drézet and Le Potier described $\mathcal{E}$ as the image of a function $\epsilon$ defined on dyadic rationals by the recursion (14). The paper's proof shows that this recursion is equivalent to the Springborn mediant rule, so the Markov fraction tree enumerates exactly the exceptional slopes. In particular, the denominators of Markov fractions are the Markov numbers, recovering Rudakov's result on ranks, and every numerator satisfies $p^2+1\equiv 0\pmod q$.

Load-bearing premise

The argument assumes that the Drézet–Le Potier classification is complete, namely that the recursion (14) really lists every slope of an exceptional bundle on $\mathbb{P}^2$; the paper shows this list coincides with the Markov fractions but does not re-prove the completeness of the classification.

Editorial extensions

If this is right

The ranks of exceptional bundles on $\mathbb{P}^2$ are Markov numbers, with a proof simpler than Rudakov's.
The Markov fraction tree is an explicit recursive enumeration of all exceptional slopes, so the Diophantine properties of Markov fractions—being the worst approximable rationals with approximation constants $\ge 1/3$—automatically describe the geometry of exceptional bundles.
The slope-counting function $\mu$ is a saltus function: it is exactly the sum of its jumps, and its derivative vanishes almost everywhere, so the set of exceptional slopes has measure-zero structure.
Given the theorem, the Frobenius unicity conjecture for Markov numbers is equivalent to the statement that every exceptional bundle on $\mathbb{P}^2$ is determined up to the natural $\mathrm{Aff}_1(\mathbb{Z})$ action by its rank.
For an exceptional bundle of rank $q$ and first Chern class $p$, one has $p^2+1\equiv 0\pmod q$; for prime Markov ranks the congruence has a unique solution up to sign, confirming the unicity conjecture in that case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implicit consequence is that Diophantine statistics of Markov numbers—prime factors, growth rates, and the density results for prime Markov numbers—become statements about which Chern classes can occur for rigid bundles on $\mathbb{P}^2$.
The equality also suggests a practical enumeration: the Springborn tree is a much simpler data structure than the dyadic recursion (14), so it could serve as an indexing scheme for exceptional slopes in computational geometry.
The saltus-function picture hints that the limiting Markov irrationalities with Lagrange number $<3$, left open by Springborn, might be realized as limits of slopes along infinite chains of exceptional bundles.
The del Pezzo analogues discussed at the end of the paper suggest a template for other surfaces: match a mutation tree for ranks to a recursion for slopes, and the exceptional-slope set may again be governed by a Markov-type Diophantine equation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves that Springborn's Markov fractions are exactly the slopes of exceptional bundles on P2. The new thing is the identification itself, plus a simpler route to Rudakov's result that exceptional ranks are Markov numbers. The proof does what it says: it takes the Drézet–Le Potier recursion (14), shows it is equivalent to the Springborn mediant rule (1) via identity (15), and then relies on the DLP classification for completeness. I rechecked the algebra around (15); it reduces correctly to the Markov equation using Lemma 2.2, and the dyadic levels are cofinal in the same tree, so the image argument is sound.

The paper is honest about its debts. It states outright that the proof can essentially be extracted from Rudakov, and the main lift is from [8]. That is not a flaw; it is accurate framing. The direct derivation of (15) is a genuine contribution and makes the correspondence transparent. The note that the DLP function and Springborn's function are two descriptions of the same set is well placed.

Soft spots are minor. The completeness of the DLP classification is external; if that classification had a gap, the main theorem would inherit it, but there is no reason to doubt such a standard result. Proposition 2.1 is terse, especially the step proving integrality of p'_1, though the argument is recoverable. There are a few typos (the duplicated q'_1 in the displayed formula in Lemma 2.2 is one). Section 4, on extensions to real numbers and the question mark function, is interesting but somewhat tangential; the McShane identity argument for Proposition 4.1 is only sketched. The final conjectures reformulate Frobenius unicity in bundle language but do not resolve it, which the paper does not claim.

The citation pattern looks fine. Self-citation appears only in Section 4 and is appropriate. The paper is a solid, checkable piece of mathematics: the central claim is well supported, the algebra is correct, and the limitations are stated rather than hidden.

This deserves a serious referee. It is not a breakthrough, but it is a useful unification that will interest both algebraic geometers and number theorists. I would accept it after minor revision, asking for a slightly expanded proof of Proposition 2.1 and cleanup of the typos.

Referee Report

0 major / 6 minor

Summary. The paper identifies the set of slopes of exceptional vector bundles on the projective plane with the Markov fractions recently defined by Boris Springborn. The author recalls the Markov fraction tree and the Springborn mediant rule, then proves the algebraic identity (15) showing that the Drézet–Le Potier recursion (14) for the exceptional-slope function on dyadic rationals is exactly the Springborn mediant rule. Theorem 3.1 then follows from the Drézet–Le Potier classification of exceptional slopes, and Rudakov's theorem that the ranks of exceptional bundles are Markov numbers is obtained as a corollary. The final sections discuss the relation to Minkowski's question-mark function, the Frobenius unicity conjecture, and possible generalizations to del Pezzo surfaces.

Significance. If correct, the main theorem provides a new and conceptually simple bridge between algebraic geometry and Diophantine approximation: the exceptional slopes on P^2 are the worst approximable rationals introduced by Springborn. The central identity (15) is derived cleanly, with the Markov equation entering exactly in the last step, and it gives a genuine simplification of Rudakov's route to the Markov-number ranks. The paper is explicit that the completeness of the Drézet–Le Potier classification is an external input, so the contribution is the identification of two existing descriptions rather than a new classification. The reformulation of the Frobenius unicity conjecture in terms of exceptional bundles is attractive and potentially useful. The presentation is terse in places and contains several typos, but no load-bearing mathematical flaw was identified.

minor comments (6)

[§3, proof of Theorem 3.1] The final step 'Now the theorem follows' is very terse: it should state explicitly that the properties ε(-x)=-ε(x) and ε(x+n)=ε(x)+n give ε(0)=0 and ε(1)=1, and that, by induction, each dyadic interval [m/2^n,(m+1)/2^n] is mapped by ε to an adjacent edge of the Markov fraction tree, so the set of all values on dyadic rationals is exactly the node set of that tree. This is an expositional clarification rather than a correction, since the argument is clear from the preceding identity.
[§3] In the sentence 'which we claim to be the set MF of Markov fractions from [0,1/2]', the notation should be MFR, or explicitly MF∩[0,1/2], since MF was defined in (5) as the full affine orbit.
[§2, Prop. 2.4] In the list of the bottom left branch, the fraction 89/233 after 408/985 is a typo: the Pell branch continues with 2378/5741, while 89/233 belongs to the Fibonacci branch. Please correct the entry.
[§2, Lemma 2.2] In the proof of Lemma 2.2, the displayed formula for q'_1 is garbled by a duplicated 'q′1 = q′1 ='; it should read q'_1 = (q_2^2+q_3^2)/(p_2q_3-p_3q_2).
[§2, Prop. 2.1] The sentence 'The relation (9) now follows from the construction of the tree' is vague; a direct reference to Markov's theorem on the Conway topograph (Fig. 3) would clarify why the triples (q_1,q_2,q_3) satisfy the Markov equation and hence the Vieta relation (7).
[§4, Prop. 4.1] The proof of Proposition 4.1 is only a reference to the McShane identity; please spell out the normalization that makes the saltus formula (21) match µ(0)=0 and µ(1)=1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Markov-fraction/exceptional-slope match is an external-import result with independent algebraic verification.

full rationale

The central theorem, Theorem 3.1, identifies the set E of exceptional slopes on P2 with Springborn's Markov fractions. The proof imports two external ingredients: the Drézet–Le Potier classification, which describes E as the image of the dyadic function ε satisfying recursion (14), and Springborn's independent definition of Markov fractions via the mediant rule (1). The paper then proves algebraically that recursion (14) coincides with the Springborn mediant rule for adjacent Markov fractions, via identity (15), using the Markov equation and the determinant relation (9). This is a genuine matching of two externally given objects, not a fitting or a renaming: no parameter is fitted to a subset of data and later called a prediction, and the Markov fraction set is defined independently of the exceptional-bundle classification. The only self-citation, [29] in Section 4, concerns the Minkowski question-mark function and Lyapunov exponents and is not used to prove the main theorem. The residual reliance on the completeness of the Drézet–Le Potier classification is a normal appeal to an established external theorem, not a circular step. No self-definitional, fitted-input, or self-citation-load-bearing reduction was found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. It relies on four external theorems, two of which are the classification results of Drézet-Le Potier and the Markov fraction construction of Springborn.

assumptions (4)

domain assumption Drézet-Le Potier classification: the set of exceptional slopes on P2 is exactly the image of the function ε on dyadic rationals defined by recursion (14).
Theorem 3.1 relies on this result from [8] to conclude that the two trees have the same vertex set.
domain assumption Springborn's bijection μ: Q∩[0,1] -> MFR in equation (3) from [30].
Used to define Markov fractions and to infer the set equality from the tree construction.
standard math Markov's theorem that all Markov triples are generated from (1,1,1) by Vieta involutions and permutations, used in Prop 2.1.
This classical theorem underpins the claim that the denominators in the Markov fraction tree are Markov numbers.
domain assumption Uniqueness of exceptional bundles by slope on P2, from [8].
The paper treats the slope as the classifying invariant, which is a theorem of Drézet-Le Potier.

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Pith. "Pith review of Markov fractions and the slopes of the exceptional bundles on $\mathbb P^2$." pith.science (2026). https://pith.science/paper/QKFUEYZY

@misc{pith2026250106779,
  author       = {Pith},
  title        = {Pith review of: Markov fractions and the slopes of the exceptional bundles on $\mathbb P^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKFUEYZY}},
  note         = {Machine review of arXiv:2501.06779}
}

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.

Figures

Figures reproduced from arXiv: 2501.06779 by the authors.

**Figure 1.** The Farey tree of fractions between 0 and 1. The Markov fraction tree is the modification of this Farey tree, where the Farey mediant is replaced by the Springborn mediant p1 q1 ∗ p2 q2 = p1q1 + p2q2 q 2 1 + q 2 2 , (1) or, in the reduced form, p1 q1 ∗ p2 q2 = p q , p = p1q1 + p2q2 p2q1 − p1q2 , q = q 2 1 + q 2 2 p2q1 − p1q2 . (2) By definition, the Markov fractions between 0 and 1/2 are defined recursively using t… view at source ↗

**Figure 2.** The Markov fraction tree with the Springborn local rule. which is a version of Frobenius parametrisation of Markov numbers [12]. By definition the function µ(x) satisfies the property µ a b ⊕ c d = µ a b ∗ µ c d , |ad − bc| = 1, (4) intertwining Farey and Sprinborn mediants of neighbours on the Farey tree. The set of all Markov fractions is defined as MF := {n ± p q , p q ∈ MF R, n ∈ Z}. (5) The reduced se… view at source ↗

**Figure 3.** Markov numbers on the Conway topograph. The numbers in three domains meeting at one vertex form Markov triples (the singular Markov triples (1,1,1) and (1,1,2) are left outside the chosen part of the full Conway topograph). Consider a part of the Markov fraction tree shown on [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Proof. The relation (8) follows from the direct calculation: p2q3 − p3q2 = p2(q 2 1 + q 2 2 ) p2q1 − p1q2 − q2(p1q1 + p2q2) p2q1 − p1q2 = p2q 2 1 − p1q1q2 p2q1 − p1q2 = q1. The last two relations are now obvious: p ′ 1 = p2q2 + p3q3 p2q3 − p3q2 = p2q2 + p3q3 q1 , q′ 1 …

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [8]

Dr` ezet and J

J.-M. Dr` ezet and J. Le PotierFibr` es stables et fibr` es exceptionnels surP2. Ann. Sci. Ecole Norm. Sup. (4) 18 (1985), 193-243

work page 1985

[2] [1]

Aigner Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Math- ematical Journey from Irrational Numbers to Perfect Matchings

M. Aigner Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Math- ematical Journey from Irrational Numbers to Perfect Matchings . Springer, 2013

work page 2013

[3] [2]

Baragar On the unicity conjecture for Markoff numbers

A. Baragar On the unicity conjecture for Markoff numbers. Canad. Math. Bull. 39:1 (1996), 3-9

work page 1996

[4] [3]

Bourgain, A.Gamburd, P

J. Bourgain, A.Gamburd, P. Sarnak Markoff surfaces and strong approximation. Compt. Rendus Math. 354:2 (2016), 131-135

work page 2016

[5] [4]

Buchstaber and A.P

V.M. Buchstaber and A.P. Veselov Conway topograph, P GL2(Z)-dynamics and two- valued groups. Russian Math. Surveys 74:3 (2019), 387-430. 11

work page 2019

[6] [5]

Cassels An Introduction to Diophantine Approximation

J.W.S. Cassels An Introduction to Diophantine Approximation. Camb. Univ. Press, 1957

work page 1957

[7] [6]

Cohn Approach to Markoff ’s minimal forms through modular functions

H. Cohn Approach to Markoff ’s minimal forms through modular functions . Annals of Math. 61 (1955), 1-12

work page 1955

[8] [7]

Conway The Sensual (Quadratic) Form , Carus Mathematical Monographs, Vol.26

J.H. Conway The Sensual (Quadratic) Form , Carus Mathematical Monographs, Vol.26. Mathematical Association of America, 1997

work page 1997