Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

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.

14 pages. v2: references updated

References in corpus (3)

• Minkowski Polynomials and Mutations
• Maximally Mutable Laurent Polynomials
• Mutations of Laurent Polynomials and Flat Families with Toric Fibers

Cited by in corpus (24)

• Maximally Mutable Laurent Polynomials
• Laurent Inversion
• On deformations of toric Fano varieties
• On toric geometry and K-stability of Fano varieties
• Mirror Symmetry and smoothing Gorenstein toric affine 3-folds
• On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities
• On K-stability of some del Pezzo surfaces of Fano index 2
• Minimality and mutation-equivalence of polygons
• Cracked Polytopes and Fano Toric Complete Intersections
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• Projecting Fanos in the mirror
• Homogeneous deformations of toric pairs
• Deformations of Dimer Models
• Smoothing Toric Fano Surfaces Using the Gross-Siebert Algorithm
• Laurent polynomials in Mirror Symmetry: why and how?
• Databases of quantum periods for Fano manifolds
• Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series
• Fano mirror periods from the Frobenius structure conjecture
• K-moduli of Fano 3-folds can have embedded points
• Polarized rigid del Pezzo surfaces in low codimension
• Modularity of Landau-Ginzburg models
• Some examples of non-smoothable Gorenstein Fano toric threefolds
• Full exceptional collections for anticanonical log del Pezzo surfaces
• Mirrors to Del Pezzo Surfaces and the Classification of $T$-Polygons