On pliability of del Pezzo fibrations and Cox rings

Ahmadinezhad, Hamid

doi:10.1515/crelle-2014-0095

Cited by 20 publications

(59 citation statements)

References 32 publications

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“…Given such an ω we can also form a simplicial fan Σ where σ ∈ Σ if and only if ω ∈ D i | i ∈ σ and Σ also depends only on the chamber that ω sits in. The following Proposition collects a few well-known facts that can easily be synthesized from the literature [12,7,13,1]: Proposition 24. Let D : Z * m → L * be a homomorphism as above, choose a chamber in the secondary fan as described above, and let ω be a stability condition in it.…”

Section: Simplicial Toric Varieties and Well-formed Complete Intersecmentioning

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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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: Simplicial Toric Varieties and Well-formed Complete Intersecmentioning

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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“…Moreover, the coordinate of degree 6 is eliminated near P a using the equation ae = F . We already know that F is the blowup of P a in P (1,2,3,4,5,10) and thus ϕ contracts S to P a on X.…”

Section: 22mentioning

confidence: 99%

“…The main point is that S is a rational surface which is not quasismooth. It has a singularity of the form (x 2 = yz) ⊂ 1 7 (1,3,4). Thus S is contracted to a nonisolated singularity on X 16 ⊂ P (1,2,3,4,5) centred at the point of Gorenstein index 2 on the ambient space.…”

Section: 22mentioning

confidence: 99%

K3 transitions and canonical 3-folds

Coughlan

2018

Bull. London Math. Soc.

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We introduce K3 transitions as a geometric approach to studying canonical 3‐folds. These transitions link different deformation classes of canonical 3‐folds via a combination of birational contractions and smoothings. As applications, we investigate some basic properties of the web of canonical 3‐folds in small codimension and give an interesting example of a singularity in codimension 4 with obstructed smoothings, similar to the famous example of the affine cone over a degree 6 del Pezzo surface.

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“…An alternative reduction procedure has been described in [1] to which the interested reader is referred. …”

Section: Q Is Reduced If and Only If For Everymentioning

confidence: 99%

“…The fact that Σ is simplicial ensures that N Σ(1) is still a full sublattice of N , and the rows of T n give a basis of N Σ (1) . Hence N/N Σ(1) ∼ = Z n /L r (T n ).…”

Section: Proof Of Theorem 24mentioning

confidence: 99%

Z-linear Gale duality and poly weighted spaces (PWS)

Rossi

Terracini

2016

Linear Algebra and its Applications

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MSC: 14M25 06D50Keywords: Q-factorial complete toric varieties Gale duality Weighted projective spaces Hermite normal form Smith normal formThe present paper is devoted to discussing Gale duality from the Z-linear algebraic point of view. This allows us to isolate the class of Q-factorial complete toric varieties whose class group is torsion free, here called poly weighted spaces (PWS). It provides an interesting generalization of weighted projective spaces (WPS).

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On pliability of del Pezzo fibrations and Cox rings

Cited by 20 publications

References 32 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

K3 transitions and canonical 3-folds

Z-linear Gale duality and poly weighted spaces (PWS)

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