Degenerations of the complex projective plane with only rational singularities

Marcos Canedo; Giancarlo Urzúa

Degenerations of the complex projective plane with only rational singularities

Algebraic Geometry 2026-07-21 v1 Geometric Topology Symplectic Geometry

Authors: Marcos Canedo , Giancarlo Urzúa

Abstract

Wahl's conjecture states that two-dimensional singularities admitting a rational homology disk smoothing are weighted homogeneous. Assuming the conjecture, we classify all normal degenerations of the complex projective plane with only rational singularities. They are precisely the surfaces classified by Manetti and Hacking--Prokhorov, which are controlled by the Markov equation, together with six new degenerations containing four non-log canonical singularities.

Keywords

degenerations complex projective plane rational singularities Wahl's conjecture Markov equation smoothing

Cite

@article{arxiv.2607.19348,
  title  = {Degenerations of the complex projective plane with only rational singularities},
  author = {Marcos Canedo and Giancarlo Urzúa},
  journal= {arXiv preprint arXiv:2607.19348},
  year   = {2026}
}