| Summary: | We first study the nearly Gorensteinness of Ehrhart rings arising from lattice polytopes. We give necessary conditions and sufficient conditions on lattice polytopes for their Ehrhart rings to be nearly Gorenstein. Using this, we give an efficient method for constructing nearly Gorenstein polytopes. Moreover, we determine the structure of nearly Gorenstein 0/1-polytopes.
Next, we introduce a new constant associated to a Fano polygon, called the PCR index, and prove that it is invariant under mutation. We obtain a convenient formula for the PCR index in terms of the lengths and heights of the edges of the polygon and apply this to show that two given Fano polygons with the same singularity content are not mutation-equivalent. Finally, we give an alternative way to classify the minimal Fano triangles which have empty basket of singularities; we accomplish this using Markov-like Diophantine equations.
Then, 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, which were recently conjectured to constitute all Kähler-Einstein Fano polygons. We show that all polygons in this subclass are minimal and that each mutation-equivalence class has at most one Fano polygon belonging to this subclass. Finally, we provide counterexamples to the aforementioned conjecture and discuss several of their properties.
Finally, we study generalised flatness constants of lattice polytopes.
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