Smoothable del Pezzo surfaces with quotient singularities

Abstract

We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to solutions of a Markov-type equation. The remaining surfaces are obtained as deformations of the toric surfaces or belong to a finite list of sporadic surfaces.

36 pages. v2: corrected attribution of Prop. 2.7, added reference

References in corpus (3)

• Compact moduli of plane curves
• Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties
• Classification of log del Pezzo surfaces of index $\le 2$

Cited by in corpus (48)

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• Mutations of Laurent Polynomials and Flat Families with Toric Fibers
• Markov numbers and Lagrangian cell complexes in the complex projective plane
• Mutations of fake weighted projective planes
• K-stability of smooth del Pezzo surfaces
• Exceptional bundles associated to degenerations of surfaces
• The maximal number of singular points on log del Pezzo surfaces
• Deformations of Nodal Kähler-Einstein Del Pezzo Surfaces with Discrete Automorphism Groups
• On toric geometry and K-stability of Fano varieties
• A note on degenerations of del Pezzo surfaces
• Smoothing singular extremal Kähler surfaces and minimal Lagrangians
• On deformations of toric Fano varieties
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• Applications of the moduli continuity method to log K-stable pairs
• Combinatorial aspects of exceptional sequences on (rational) surfaces
• Moduli of cubic surfaces and their anticanonical divisors
• Del Pezzo Surfaces in Weighted Projective Spaces
• Singularity content
• The dualizing sheaf on first-order deformations of toric surface singularities
• Minimality and mutation-equivalence of polygons
• Projecting Fanos in the mirror
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• Infinitely many monotone Lagrangian tori in del Pezzo surfaces
• Surgeries on torus knots, rational balls, and cabling
• Log canonical degenerations of del Pezzo surfaces in Q-Gorenstein families
• Laurent polynomials in Mirror Symmetry: why and how?
• Singular symplectic spaces and holomorphic membranes
• Weak Approximation for Low Degree del Pezzo Surfaces
• Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series
• Moduli of surfaces in ${\mathbb{P}}^3$
• Mutations of Fake Weighted Projective Spaces
• On certain quantifications of Gromov's non-squeezing theorem
• K-moduli of Fano 3-folds can have embedded points
• Jordan constant for Cremona group of rank 3
• Quasi-period collapse for duals to Fano polygons: an explanation arising from algebraic geometry
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• Geography of minimal surfaces of general type with ${\mathbb{Z}}_2^2$-actions and the locus of Gorenstein stable surfaces
• Horizontal decompositions, II
• Deformations of weak $\mathbb{Q}$-Fano 3-folds