| Summary: | Mirror symmetry evokes a correspondence between deformation equivalence classes of toric varieties and mutation equivalence classes of the corresponding Fano varieties. This thesis discusses many computations and examples of this ilk, in the case when the varieties are 2-dimensional and permitted to possess cyclic quotient singularities.
The mutation graph of weighted projective planes has been well studied by Akhtar-Kasprzyk. We similarly analyse the mutation graph of P1 x P1 which involves looking at quivers and Plucker coordinates.
An algorithm is presented to classify mutation equivalence classes of Fano polygons where the corresponding surfaces have fixed singularities, These surfaces are subsequently studied using Laurent inversion and found to lie in a cascade structure introduced by Reid-Suzuki.
By studying the combinatorics of Fano polygons, which involves matrix calculcations, continued fractions and r-modular sequences, we provide results regarding combinatorics of cyclic quotient singularities that do not occur for a del Pezzo surface admitting a toric degeneration.
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