Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

We classify non-smooth del Pezzo surfaces with 1 3 (1, 1) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake [14]), we tabulate their biregular invariants, we give good model constructions for surfaces in all families as degeneracy loci in rep quotient varieties, and we prove that precisely 26 families admit qG-degenerations to toric surfaces. This work is part of a program to study mirror symmetry for orbifold del Pezzo surfaces [2].In this paper we:(I) Classify non-smooth del Pezzo surfaces with 1 3 (1, 1) points in precisely 29 qG-deformation families. We further structure the classification into six unprojection cascades, determine their biregular invariants and their directed MMP together with a distinguished configuration of negative curves on the minimal resolution. This overlaps with work of Fujita and Yasutake [14].(II) Construct good models for surfaces in all families as degeneracy loci in rep quotient varieties. In all but two cases, the rep quotient variety is a simplicial toric variety.(III) Prove that precisely 26 of the 29 families admit a qG-degeneration to a toric surface.The classification is summarised in table 1 and table 2, which also plot invariants and provide good model constructions of surfaces in all families.This work is part of a program to understand mirror symmetry for orbifold del Pezzo surfaces [2,16,26,31,27,10] and it is aimed specifically at giving evidence for the conjectures made in [2].The rest of the introduction is organised as follows: in § 1.1 we give precise statements of our main results; in § 1.2 we say a few words about the context of [2]; in § 1.3 we outline the structure of the paper. Our results1.1.1 The classification and its cascade structuren acts linearly on C 2 with weights a, b ∈ 1 n Z /Z. We always assume no stabilisers in codimension 0, 1, that is, hcf(a, n) = hcf(b, n) = 1. A del Pezzo surface is a surface X with cyclic quotient singularities and −K X ample. The Fano index of X is the largest positive integer f > 0 such that −K X = f A in the Class group Cl X.Remark 2. In this paper we view a del Pezzo surface X with quotient singularities as a variety. Such a surface is in a natural way a smooth orbifold (or DM stack), but we mostly ignore this structure. Thus for us Cl X is the Class group of Weil divisors on X modulo linear equivalence. In particular, although K X is a Cartier divisor on the orbifold, we think of it as a Q-Cartier divisor on the underlying variety (the coarse moduli space of the orbifold) and then to say that it is ample is to say that a positive integer multiple is Cartier and ample. See [2] for a discussion of qG-deformations of del Pezzo surfaces with cyclic quotient singularities. In particular, it is explained there that the singularity 1 3 (1, 1) is qG-rigid and the degree d = K 2 is locally constant in qG-families.We classify qG-deformation families of del Pezzo surfaces with k ≥ 1 1 3 (1, 1) points. It follows for example from the proof of [2, Lemma 6] tha...

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confidence: 99%

Del Pezzo surfaces with $$\frac{1}{3}(1,1)$$ 1 3 ( 1 , 1 ) points

Corti

Heuberger

2016

manuscripta math.

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“…Thus, by (4) and (7), the v i are primitive lattice vectors in N . These vectors are all distinct: (2). Similarly v 1 , v 2 are both distinct from v 3 because their y-coordinates have different signs: y 1 = y 2 = −ℓ 1 , which is negative by (2), while y 3 is positive by (3).…”

Section: Characterization Of Polygonal Quiversmentioning

confidence: 99%

“…These vectors are all distinct: (2). Similarly v 1 , v 2 are both distinct from v 3 because their y-coordinates have different signs: y 1 = y 2 = −ℓ 1 , which is negative by (2), while y 3 is positive by (3). This observation also shows that v 1 , v 2 both lie on the line {(x, y) | y = −ℓ 1 } ⊂ N Q , but this line does not contain v 3 .…”

Section: Characterization Of Polygonal Quiversmentioning

confidence: 99%

Polygonal Quivers

Akhtar¹

2019

Preprint

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We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano polygon defines a point on a certain family of algebraic hypersurfaces. Our quivers admit a generalized mutation which preserves balancing and coincides with combinatorial mutation of Fano polygons whenever both operations are defined. We characterize balanced quivers arising from Fano polygons and discuss generalizations to higher dimensions.

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“…Example 3.18. Consider the toric surface (using the notation for these surfaces appearing in [3]). X 5,5/3 associated with the Fano polygon shown below.…”

Section: Mutations Of Polytopesmentioning

confidence: 99%

“…In particular we see that finite mutation classes of polygons fall into four types A n 1 , for n ∈ Z ≥0 , A 2 , A 3 , and D 4 . There is a close connection between mutation classes of Fano polygons and Q-Gorenstein deformations of the corresponding toric varieties which is described in detail in [3]. Following these ideas we predict the existence of a finite type parameter space for these deformations, together with a boundary stratification such that each zero stratum corresponds to a polygon in the given mutation class, and the 1-strata corresponds to the mutation families constructed by Ilten [26].…”

Section: Introductionmentioning

confidence: 97%