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The birational geometry of Markov numbers

Giancarlo Urzúa, Juan Pablo Zúñiga

TL;DR

The paper advances the birational classification of degenerations of ${\mathbb P}^2$ to quotient-singularity surfaces by showing that all Markov-number controlled degenerations are connected through explicit MMP sequences. It develops a rich combinatorial and geometric framework based on Hirzebruch–Jung continued fractions, Wahl chains, and Mori trains to track flips and antiflips across the Markov tree, including the Fibonacci and Pell branches. Key contributions include a sharp bound on the number of flips needed to reach a smooth deformation from any Markov triple, new equivalences and data related to the Markov conjecture, and a detailed description of the complete MMP for general Markov triples. The work deepens connections between singularities, extremal P-resolutions, Dedekind sums, and the geometry of moduli spaces of surfaces, with potential implications for Frobenius-type uniqueness questions and the structure of Markov-related moduli problems.

Abstract

It is known that all degenerations of the complex projective plane into a surface with only quotient singularities are controlled by the positive integer solutions $(a,b,c)$ of the Markov equation $$x^2+y^2+z^2=3xyz.$$ It turns out that these degenerations are all connected through finite sequences of other simpler degenerations by means of birational geometry. In this paper, we explicitly describe these birational sequences and show how they are bridged among all Markov solutions. For a given Markov triple $(a,b,c)$, the number of birational modifications depends on the number of branches that it needs to cross in the Markov tree to reach the Fibonacci branch. We show that each of these branches corresponds exactly to a Mori train of the flipping universal family of a particular cyclic quotient singularity defined by $(a,b,c)$. As a byproduct, we obtain new numerical/combinatorial data for each Markov number, and new connections with the Markov conjecture (Frobenius Uniqueness Conjecture), which rely on Hirzebruch-Jung continued fractions of Wahl singularities.

The birational geometry of Markov numbers

TL;DR

The paper advances the birational classification of degenerations of to quotient-singularity surfaces by showing that all Markov-number controlled degenerations are connected through explicit MMP sequences. It develops a rich combinatorial and geometric framework based on Hirzebruch–Jung continued fractions, Wahl chains, and Mori trains to track flips and antiflips across the Markov tree, including the Fibonacci and Pell branches. Key contributions include a sharp bound on the number of flips needed to reach a smooth deformation from any Markov triple, new equivalences and data related to the Markov conjecture, and a detailed description of the complete MMP for general Markov triples. The work deepens connections between singularities, extremal P-resolutions, Dedekind sums, and the geometry of moduli spaces of surfaces, with potential implications for Frobenius-type uniqueness questions and the structure of Markov-related moduli problems.

Abstract

It is known that all degenerations of the complex projective plane into a surface with only quotient singularities are controlled by the positive integer solutions of the Markov equation It turns out that these degenerations are all connected through finite sequences of other simpler degenerations by means of birational geometry. In this paper, we explicitly describe these birational sequences and show how they are bridged among all Markov solutions. For a given Markov triple , the number of birational modifications depends on the number of branches that it needs to cross in the Markov tree to reach the Fibonacci branch. We show that each of these branches corresponds exactly to a Mori train of the flipping universal family of a particular cyclic quotient singularity defined by . As a byproduct, we obtain new numerical/combinatorial data for each Markov number, and new connections with the Markov conjecture (Frobenius Uniqueness Conjecture), which rely on Hirzebruch-Jung continued fractions of Wahl singularities.
Paper Structure (13 sections, 22 theorems, 154 equations, 4 figures)

This paper contains 13 sections, 22 theorems, 154 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hirzebruch-Jung continued fractions and Wahl chains
  3. Markov numbers and HJ continued fractions
  4. Some characterizations of Markov numbers
  5. Birational geometry and Mori trains
  6. Degenerations of ${\mathbb{P}}^2$ as smooth deformations of ${\mathbb{F}}_1$
  7. Mutations, Mori trains, and the complete MMP
  8. MMP on the Fibonacci branch
  9. MMP on the Pell branch
  10. MMP on the branches of $(1=a<b<c)$
  11. MMP for general Markov triples
  12. The complete MMP on an arbitrary Markov triple
  13. Computations for the MMP on $(1=a<b<c)$

Key Result

Theorem 1.5

Let $(a,b<c)$ be a Markov triple, and let $i>0$. Then the MMP on corresponding to each of the two branches defined by $(a,b<c)$ stabilizes at the $3$rd flip (see Definition stabilization). Moreover, for each fixed branch the antiflips at the 3rd flip are k2A neighborhoods of one Mori train over the cyclic quotient singularity $\frac{1}{\Delta}(1,\Omega)$, where $\zeta=w_c,c-w_c$ (one value for e

Figures (4)

  • Figure 1: Markov tree and some branches
  • Figure 2: Chain of Wahl singularities
  • Figure 3: Guiding diagram for Lemma \ref{['lemma8']}
  • Figure 4: Guiding diagram for Lemma \ref{['lemma12']}

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9: Proposition \ref{['markovrc']}
  • Theorem 1.10: Proposition \ref{['combi1']}
  • ...and 53 more