Classification of terminal simplicial reflexive d-polytopes with 3d − 1 vertices

Øbro, Mikkel

doi:10.1007/s00229-007-0133-z

Cited by 7 publications

(23 citation statements)

References 16 publications

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“…Moreover, our result extends results of Øbro who classified the polytopes of the named kind with 3d − 1 vertices [20]. Our proof employs techniques similar to those used by Øbro [20] and Nill and Øbro [18], but requires more organization since a greater variety of possibilities occurs. One benefit of our approach is that it suggests a certain general pattern emerging, and we state this as Conjecture 9 below.…”

Section: Introductionsupporting

confidence: 71%

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Smooth Fano Polytopes with Many Vertices

Assarf

Joswig

Paffenholz

2014

Discrete Comput Geom

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Section: Introductionsupporting

confidence: 71%

“…To see this look at the del Pezzo polytope DP (4) Notice that φ(v) = 0 implies that the vertex opp(F, v) = φ(v) − v is contained in the set V (F, 0). The following partial converse slightly strengthens [20,Lem. 6], but the proof is essentially the same.…”

Section: 3supporting

confidence: 59%

Smooth Fano Polytopes with Many Vertices

Assarf

Joswig

Paffenholz

2014

Discrete Comput Geom

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“…Note that in this case necessarily u F , s P ≥ 0. Special facets were the crucial tool in Øbro's classification algorithm for smooth Fano polytopes [21].…”

Section: Neighboring Facets and Opposite Verticesmentioning

confidence: 99%

A bound for the splitting of smooth Fano polytopes with many vertices

Assarf

Nill

2015

J Algebr Comb

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A b s t r ac t . The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least 3d − 2 vertices are completely known. The main result of this paper deals with the case of 3d − k vertices for k fixed and d large. It implies that there is only a finite number of isomorphism classes of toric Fano d-folds X (for arbitrary d) with Picard number 2d − k such that X is not a product of a lower-dimensional toric Fano manifold and the projective plane blown up in three torus-invariant points. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.1. I n t ro d u c t i o n a n d m a i n r e s u lt s Let us first recall the basic definitions. We refer to [19,12] for more background. Let N ∼ = Z d be a lattice with associated real vector space N R := N ⊗ Z R isomorphic to R d . A polytope P is a convex, compact set in N R , its 0-dimensional faces are called vertices, and its faces of codimension 1 are called facets. If every facet F (of dimension d − 1) of a d-dimensional polytope P has exactly d vertices (i.e., F is a simplex), then P is called simplicial. The polytope P is called a lattice polytope if its vertices are lattice points (i.e., elements of N ).P is a lattice polytope, and P is full-dimensional and contains the origin 0 as an interior point, and for each facet F of P , the vertex set Vert F is a lattice basis of N . (ii) Two smooth Fano polytopes are lattice equivalent, if their vertex sets are in bijection by an affine-linear lattice automorphism.Remark 2. We decided to keep the notion of a smooth Fano polytope in order to be consistent with existing literature. However, we remark that there exists also the definition of a smooth polytope as a lattice polytope with unimodular vertex cones. A smooth Fano polytope is not a smooth polytope (but its dual polytope is).Note that any smooth Fano polytope P is necessarily simplicial. In each dimension there exist only finitely many smooth Fano polytopes up to lattice equivalence (we refer to the survey [19]). In 2007, Øbro described an explicit classification algorithm

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“…Smooth Fano dpolytopes have been intensively studied during the last decades and completely classified up to dimension 4 ([1] and [23]). In higher dimension, they are classified under some additional assumptions; for instance, when the polytopes have few vertices (see [17]), maximal number of vertices (see [5] and [20]) or some extra symmetries (see [6]).…”

Section: Toric Varieties and Frobenius Splittingmentioning

confidence: 99%