Classification of Minimal Polygons with Specified Singularity Content

Cavey, Daniel; Kutas, Edwin

doi:10.48550/arxiv.1703.05266

Cited by 3 publications

(9 citation statements)

References 12 publications

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“…It is an interesting problem to complete classifications of Fano polygons by singularity content, and some such results are available [5,6,12]. In this vain the first main result of this paper is as follows: Theorem 2.4.…”

Section: Introductionmentioning

confidence: 92%

Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon

Cavey¹,

Higashitani²

2019

Preprint

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By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose basket of residual singularities is of the form 1 r (1, s1), 1 r (1, s2), . . . , 1 r (1, s k ) for k, r ∈ Z>0, and 1 ≤ si < r is coprime to r.

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Section: Introductionmentioning

confidence: 92%

Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon

Cavey¹,

Higashitani²

2019

Preprint

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“…The toric varieties to which such a surface can degenerate are classified in [9]; applying Laurent inversion to these cases gives models for these surfaces. The main results of [9] show that either such a surface contains a single 1 k (1, 1) singularity, for k ∈ {3, 5, 6}, or is one of three exceptional cases. In this section we show that all of these surfaces are hypersurfaces in weighted projective spaces.…”

Section: Surfaces With Larger Basketsmentioning

confidence: 99%

“…In this section we show that all of these surfaces are hypersurfaces in weighted projective spaces. In particular, consider polygons 1.13 and 1.14 from [9]. While we use Laurent inversion here we could also use the Ehrhart series of the dual polygons to those appearing in [9] to guess the hypersurface model.…”

Section: Surfaces With Larger Basketsmentioning

confidence: 99%

“…In particular, consider polygons 1.13 and 1.14 from [9]. While we use Laurent inversion here we could also use the Ehrhart series of the dual polygons to those appearing in [9] to guess the hypersurface model. Polygon 1.13 is given by conv (−1, 1), (1, 1), (5, −1), (−5, −1) .…”

Section: Surfaces With Larger Basketsmentioning

confidence: 99%

“…In [9] a classification of all toric del Pezzo surfaces with a Q-Gorenstein deformation to a del Pezzo surface with only combinations of 1 3 (1, 1), 1 5 (1, 1), and 1 6 (1, 1) singularities, as listed in Theorem 1.3, is given. Theorem 1.1 tells us that there are no additional del Pezzo surfaces; that is, all such del Pezzo surfaces admit a toric degeneration to one of the toric varieties in [9]. These toric degenerations are also embedded in toric varieties with codimension ≤ 2.…”

Section: Introductionmentioning

confidence: 99%

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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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Polarized rigid del Pezzo surfaces in low codimension

Qureshi

2020

Preprint

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We classify wellformed and quasismooth polarized del Pezzo surfaces having basket of rigid orbifold points of type ki × 1 r i (1, a) : 3 ≤ ri ≤ 10, ki ∈ Z ≥0 , such that their images under their anti-canonical embedding in a weighted projective space can be described by using one of the following sets of equations: a single equation, two linearly independent equations, maximal Pfaffians of 5 × 5 skew symmetric matrix and 2 × 2 minors of size 3 square matrix. The classification is complete under certain carefully chosen bounds on the weights of ambient weighted projective spaces and it is largely based on detailed computer assisted searches by using the computer algebra system magma.

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Classification of Minimal Polygons with Specified Singularity Content

Cited by 3 publications

References 12 publications

Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon

Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon

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

Polarized rigid del Pezzo surfaces in low codimension

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