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Classification of Horikawa surfaces with T-singularities

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arxiv 2410.02943 v3 pith:BGW5CPRU submitted 2024-10-03 math.AG math.DGmath.GTmath.SG

Vicente Monreal , Jaime Negrete , Giancarlo Urzúa This is my paper · ORCID
classification math.AGmath.DGmath.GTmath.SG
keywords surfacesonlyt-singularitieshorikawaclassificationclassifyksbasmoothable
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abstract

We classify all projective surfaces with only T-singularities, ample canonical class, and $K^2=2p_g-4$. In this way, we identify all surfaces, smoothable or not, with only T-singularities in the Koll\'ar--Shepherd-Barron--Alexeev (KSBA) moduli space of Horikawa surfaces. We also prove that they are not smoothable when $p_g \geq 10$, except for the Lee-Park (Fintushel-Stern) examples, which we show to have only one deformation type unless $p_g=6$ (in which case they have two). This demonstrates that the challenging Horikawa problem cannot be addressed through complex T-degenerations. We propose new questions regarding diffeomorphism types based on our classification. Furthermore, the techniques developed in this paper enable us to classify all KSBA surfaces with only T-singularities and $K^2\leq 2p_g-3$, for example, quintic surfaces and I-surfaces.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Pseudo-hyperbolicity of Horikawa surfaces

    math.AG 2026-08 conditional novelty 7.0 of 10

    Very general Horikawa surfaces with p_g ≥ 5 (and first-kind with p_g = 3, 4) contain only finitely many rational or elliptic curves, with explicit counts in most strata.

  2. Degenerations of the complex projective plane with only rational singularities

    math.AG 2026-07 conditional novelty 7.0 of 10

    Assuming Wahl's conjecture, every normal degeneration of the projective plane to a surface with only rational singularities is one of the Markov-equation family or one of six newly found surfaces.

  3. Normal stable degenerations of Noether-Horikawa surfaces

    math.AG 2025-07 conditional novelty 7.0 of 10

    Every Q-Gorenstein smoothable normal stable Horikawa surface falls into one of seven explicit families, and its global smoothability is controlled by a single local condition at its elliptic double cone singularities.

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