Some examples of non-smoothable Gorenstein Fano toric threefolds

Petracci, Andrea

doi:10.48550/arxiv.1804.07960

Cited by 2 publications

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“…We describe an important class of non-examples. Consider the reflexive polytope P with PALP id 15, whose corresponding toric variety X P is shown to be non-smoothable by Petracci in [21]. A neighbourhood of Sing X P is isomorphic to a bundle of A 1 (surface) singularities over P 1 .…”

Section: Examplesmentioning

confidence: 99%

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Cracked polytopes and Fano toric complete intersections

Prince

2019

manuscripta math.

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We introduce the notion of cracked polytope, and -making use of joint work with Coates and Kasprzyk -construct the associated toric variety X as a subvariety of a smooth toric variety Y under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of X to exist inside Y . We exhibit a relative anti-canonical divisor for this smoothing of X, and show that the general member is simple normal crossings.Conventions. Throughout this article N ∼ = Z n will refer to an n-dimensional lattice, and M := hom(N, Z) will refer to the dual lattice. Given a ring R we write N R := N ⊗ Z R and M R := M ⊗ Z R. For brevity we let [k] denote the set {1, . . . , k} for each k ∈ Z ≥1 . We work over an algebraically closed field k of characteristic zero. Cracked polytopesIn this section we introduce the notion of cracked polytope, which will form our main object of study, and, in the Fano setting, characterize its dual polytope. We will assume basic ideas and results from toric geometry -see [9, 10] -throughout.

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Section: Examplesmentioning

confidence: 99%

“…The toric variety X s ∼ = F 1 with (−1)-curve E, and D s = 2E; hence S cannot be positive. The same analysis applies to the reflexive polytopes with PALP ids in the set (a subset of the list appearing in [21]), {16, 58, 59, 61, 65, 66, 192, 193, 197}.…”

Section: Examplesmentioning

confidence: 99%

Cracked polytopes and Fano toric complete intersections

Prince

2019

manuscripta math.

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“…The deformation theory of toric varieties is lately an active research area, see [7], [8], [9], [5], [6], [13], [14]. One of the main motivations comes from the classification problem of smooth Fano varieties, cf.…”

Section: Introductionmentioning

confidence: 99%

On deformations of affine Gorenstein toric varieties

Filip¹

2020

Preprint

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Some examples of non-smoothable Gorenstein Fano toric threefolds

Cited by 2 publications

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Cracked polytopes and Fano toric complete intersections

Cracked polytopes and Fano toric complete intersections

On deformations of affine Gorenstein toric varieties

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