Classification of Horikawa surfaces with T-singularities

Abstract: We classify all projective surfaces with only T-singularities, $K^2=2p_g-4$, and ample canonical class. 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, and 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 $K^2=2p_g-3$, e.g. quintic surfaces and I-surfaces.