Classification of Horikawa surfaces with T-singularities

Vicente Monreal; Jaime Negrete; Giancarlo Urzúa

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arXiv · 2410.02943

Classification of Horikawa surfaces with T-singularities

Vicente Monreal·Jaime Negrete·Giancarlo Urzúa

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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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Vicente Monreal, Jaime Negrete, Giancarlo Urzúa. 2024-10-03. Classification of Horikawa surfaces with T-singularities. https://arxiv.org/abs/2410.02943

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