On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

Oneto, Alessandro; Petracci, Andrea

doi:10.1515/advgeom-2017-0048

Cited by 10 publications

(15 citation statements)

References 35 publications

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“…We put our results in the context of a general program to understand mirror symmetry for orbifold del Pezzo surfaces [2,16,26,31,27,10] and answer the question: Why classify del Pezzo surfaces with 1 3 (1, 1) points? qG-deformations of surface singularities is a technical notion that ensures that the canonical class is wellbehaved in families: in particular, K 2 is locally constant in a qG-family of proper surfaces.…”

Section: Contextmentioning

confidence: 99%

“…Conditions (a) and (b) are of course also natural from a purely classification-theoretic perspective. Paper [26] computes (part of) the quantum orbifold cohomology of our surfaces.…”

Section: Example 14 (A) a Singularity Is Of Class T If And Only If Imentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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].…”

mentioning

confidence: 99%

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

Corti

Heuberger

2016

manuscripta math.

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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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Section: Contextmentioning

confidence: 99%

“…Conditions (a) and (b) are of course also natural from a purely classification-theoretic perspective. Paper [26] computes (part of) the quantum orbifold cohomology of our surfaces.…”

Section: Example 14 (A) a Singularity Is Of Class T If And Only If Imentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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

Corti

Heuberger

2016

manuscripta math.

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“…We tabulate those quivers Q ′ present case the single 1 k (1, 1) singularity). The parameters which appear are related to the orbifold Quantum cohomology of X (l) k and were studied by Oneto-Petracci [27] when k = 3. Note that, as well as its intrinsic interest, a cluster algebra description of the surfaces X (l) k provides deep geometric insights.…”

Section: Mutation Classes Of Polygonsmentioning

confidence: 99%

Del Pezzo surfaces with a single $1/k(1,1)$ singularity

Cavey¹,

Prince²

2020

J. Math. Soc. Japan

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Inspired by the recent progress by Coates-Corti-Kasprzyk et al. on Mirror Symmetry for del Pezzo surfaces, we show that for any positive integer k the deformation families of del Pezzo surfaces with a single 1 k (1, 1) singularity (and no other singular points) fit into a single cascade. Additionally we construct models and toric degenerations of these surfaces embedded in toric varieties in codimension ≤ 2. Several of these directly generalise constructions of Reid-Suzuki (in the case k = 3). We identify a root system in the Picard lattice, and in light of the work of Gross-Hacking-Keel, comment on Mirror Symmetry for each of these surfaces. Finally we classify all del Pezzo surfaces with certain combinations of 1 k (1, 1) singularities for k = 3, 5, 6 which admit a toric degeneration.DEL PEZZO SURFACES WITH A SINGLE 1 k (1, 1) SINGULARITY 3 du Val in the cases k = 1, 2, which are smooth and ordinary double points respectively. These are the only two cases for which the singularity 1 k (1, 1) is canonical. Definition 2.1. Given an arbitrary quotient singularity σ = 1 R (a, b), set k = gcd(a + b, R), c = (a + b)/k and r = R/k. Then σ can be written in the form 1 kr (1, kc − 1) and:Definition 2.1 is motivated by the work of Wahl [30] and Kollár-Shepherd-Barron [29] on the deformations of singularities. Discussion of these definitions from a toric viewpoint can be found in Akhtar-Kasprzyk [3]. A cyclic quotient singularity is a T -singularity if and only if it admits a Q-Gorenstein smoothing. Alternatively an R-singularity is rigid under any Q-Gorenstein deformation.Example 2.2. The singularities 1 k (1, 1) are R-singularities precisely when k = 3 or k ≥ 5. The singularities 1 2 (1, 1) and 1 4 (1, 1) are T -singularities. An algebraic surface is Q-Gorenstein if it is normal and the canonical divisor class is Q-Cartier.

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A 1-Dimensional Component of K-Moduli of del Pezzo Surfaces

Petracci

2023

Springer Proceedings in Mathematics &Amp; Statistics

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On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

Cited by 10 publications

References 35 publications

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

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

Del Pezzo surfaces with a single $1/k(1,1)$ singularity

A 1-Dimensional Component of K-Moduli of del Pezzo Surfaces

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