2024/12/25 by Joaquín Moraga, Moraga, Joaquín, José Ignacio Yáñez +1 · 1 citation
Mathematics · #14D10 #14J17 #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Graph theory and applications #Primary: 14E30 #Secondary: 14M25
paper · pdf · doi:10.48550/arxiv.2412.18830
openalex publication_date 2024/12/25 · openalex created_date 2024/12/31 · openalex updated_date 2026/07/28
A Calabi-Yau pair of index one and complexity zero is toric. Furthermore, a Calabi-Yau pair of index one and complexity one is of cluster type. In this article, we study Calabi-Yau pairs of index one and complexity two. We develop machinery to decide whether a Calabi-Yau of complexity two is of cluster type. This approach reduces the problem to studying del Pezzo fibrations over toric varieties. We apply this to the setting of Gorenstein del Pezzo surfaces of Picard rank one. We prove that such a surface X is cluster type if and only if X has only A-type singularities and either vol(X)>1 or |X\rm sing|≤ 3.