Vicente Monreal | Searcharxiv

arXiv2026

Functorial stratifications of singularities in characteristic 0

We prove that the riso-stratifications of singularities of affine algebraic sets introduced by Bradley-Williams--Halupczok are embedding independent and can be computed \'etale locally. Building on this, we upgrade their procedure, originally formulated in non-archimedean and model-theoretic language, into a sharp canonical and functorial stratification process for schemes of finite type over fields of characteristic 0. Our work enables comparisons with classical approaches to singularities and makes these stratifications available without requiring familiarity with the original methods.

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arXiv2024

Classification of Horikawa surfaces with T-singularities

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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