Moscow Mathematical Journal

Volume 25, Issue 2, April–June 2025 pp. 197–248.

The Birational Geometry of Markov Numbers

Authors: Giancarlo Urzúa (1) and Juan Pablo Zúñiga (2)
Author institution:(1) Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile.
(2) Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile.

Summary:

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.

2020 Math. Subj. Class. 14E30, 14D06, 32S30

Keywords: Markov numbers, MMP, Wahl singularities, birational geometry, flips.

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