Abstract
We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.
Highlights
We explore mirror symmetry for del Pezzo surfaces with cyclic quotient singularities
The canonical class of X is a Q-Cartier divisor and it makes sense to say that X is a del Pezzo surface, that is, that the anti-canonical divisor −KX is ample
In our formulation below, one side of mirror symmetry consists of the set of qG-deformation classes of locally qG-rigid del Pezzo surfaces, that is, of del Pezzo surfaces with residual singularities
Summary
Consider a del Pezzo surface X with isolated cyclic quotient singularities. X is analytically locally (oretale locally if you prefer) isomorphic to a quotient C2/μn, where without loss of generality μn acts with weights (1, q) with hcf(q, n) = 1.
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