Tag: Giancarlo Urzua

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.

We defined wormholes in https://arxiv.org/abs/2102.02177 (joint with Nicolás Vilches). Conjecturally it is a way to non-continuous travel in the KSBA compactification of the moduli space of surfaces of general type. It depends on a particular MMP. In that paper, we verified the conjecture in several cases, but many remain open. Beyond the certainty of the conjecture, it would be interesting to know about changes in the topology or differential structure after traveling through a wormhole. In this talk, I will exemplify what we know, and I will state open questions, which also include a mysterious combinatorial invariant delta that remains constant in this journey and seems to be part of some particular sequence of integers.