Mirror Symmetry and the Classification of Orbifold Del Pezzo Surfaces

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. We explore mirror symmetry for del Pezzo surfaces with cyclic quotient singularities. We begin by stating two logically independent conjectures. In Conjecture A we try to imagine what consequences mirror symmetry may have for classification theory. In Conjecture B we make what we mean by mirror symmetry precise. This work owes a great deal to conversations with Sergey Galkin, and to the pioneering papers by Galkin–Usnich [16] and Gross–Hacking–Keel [19, 20].