Marcos Canedo | Searcharxiv

arXiv2026

Degenerations of the complex projective plane with only rational singularities

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.

math.AG↗

arXiv2026

Rational homology disk degenerations of elliptic surfaces

In this paper, a $\mathbb{Q}$HD singularity is a weighted homogeneous normal surface singularity admitting a rational homology disk ($\mathbb{Q}$HD) smoothing. These singularities are rational but often not log canonical. We classify all $\mathbb{Q}$HD degenerations of nonsingular projective elliptic surfaces, extending Kawamata's classification of the case with only Wahl singularities (i.e., log terminal $\mathbb{Q}$HD singularities). We also realize all $\mathbb{Q}$HD degenerations of Dolgachev surfaces $D_{a,b}$ with one $\mathbb{Q}$HD singularity, for every pair of integers $a,b$. For each such degeneration, we construct a minimal semi log canonical (slc) birational model via a Seifert partial resolution in the sense of Wahl followed by semistable flips. Finally, we prove that these minimal slc models are unobstructed and deform to the recent degenerations of Dolgachev surfaces constructed by D. Lee and Y. Lee.

math.AG↗