Giancarlo Urzua, The Birational Geometry of Markov Numbers

The projective plane is rigid. However, it may degenerate to surfaces with quotient singularities. After the work of Bădescu and Manetti, Hacking and Prokhorov 2010 classified these degenerations completely. They are Q-Gorenstein partial smoothings of P(a², b², c²), where a, b, c satisfy the Markov equation x²+y²+z²=3xyz. Let us call the corresponding degenerations Markovian planes. They are part of a bigger picture of degenerations with Wahl singularities, where there is an explicit MMP whose final results are either K nef, smooth deformations of ruled surfaces, or Markovian planes. Although it is a final result of MMP, we can nevertheless run MMP on small modifications of Markovian planes to obtain new numerical/combinatorial data for Markov numbers via birational geometry. New connections with Markov conjecture (i.e. Frobenius Uniqueness Conjecture) are byproducts. This is joint work with Juan Pablo Zúñiga (Ph.D. student at UC Chile), the pre-print can be found here https://arxiv.org/abs/2310.17957.