2024/02/05 by T. E. Hall, Hall, Thomas
Computer Science · #14J45 (Secondary) #52B99 (Primary) #Algebraic Geometry (math.AG) #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2402.02832
openalex publication_date 2024/02/05 · openalex created_date 2024/02/08 · openalex updated_date 2026/07/28
We study a subclass of Kähler-Einstein Fano polygons and how they behave under mutation. The polygons of interest are Kähler-Einstein Fano triangles and symmetric Fano polygons. In particular, we find an explicit bound for the number of these polygons in an arbitrary mutation-equivalence class. An important mutation-invariant of a Fano polygon is its singularity content. We extend the notion of singularity content and prove that it is still a mutation-invariant. We use this to show that if two symmetric Fano polygons are mutation-equivalent, then they are isomorphic. We further show that if two Kähler-Einstein Fano triangles are mutation-equivalent, then they are isomorphic. Finally, we show that if a symmetric Fano polygon is mutation-equivalent to a Kähler-Einstein triangle, then they are isomorphic. Thus, each mutation-equivalence class has at most one Fano polygon which is either a Kähler-Einstein triangle or symmetric. A recent conjecture states that all Kähler-Einstein Fano polygons are either triangles or are symmetric. We provide a counterexample P to this conjecture and discuss several of its properties. For instance, we compute iterated barycentric transformations of P and find that (a) the Kähler-Einstein property is not preserved by the barycentric transformation, and (b) P is of strict type B2. Finally, we find examples of Kähler-Einstein Fano polygons which are not minimal.