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33
An algorithm for the classification of smooth Fano polytopes
"... We present an algorithm that produces the classification list of smooth Fano dpolytopes for any given d ≥ 1. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano dpolytopes for d ≤ 7. There are 7622 isomorphism classes of ..."
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We present an algorithm that produces the classification list of smooth Fano dpolytopes for any given d ≥ 1. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano dpolytopes for d ≤ 7. There are 7622 isomorphism classes of smooth Fano 6polytopes and 72256 isomorphism classes of smooth Fano 7polytopes. 1
The number of vertices of a Fano polytope
, 2004
"... Let X be a smooth, complex Fano variety. The Picard group Pic X is a finitely generated abelian group. The Picard number ρX is the rank of PicX, and coincides with the second Betti number of X. The pseudoindex of X was introduced in [20] as ιX: = min{−KX · C  C rational curve in X} ∈ Z>0. The ..."
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Cited by 23 (1 self)
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Let X be a smooth, complex Fano variety. The Picard group Pic X is a finitely generated abelian group. The Picard number ρX is the rank of PicX, and coincides with the second Betti number of X. The pseudoindex of X was introduced in [20] as ιX: = min{−KX · C  C rational curve in X} ∈ Z>0. The object of this paper is the following: Theorem. Let X be a smooth toric Fano variety of dimension n, Picard number ρX and pseudoindex ιX. Then: (i) ρX(ιX − 1) ≤ n, with equality if and only if X ∼ = (P ιX−1 ρX (ii) ρX ≤ 2n, with equality if and only if n is even and X ∼ = (S3) n 2, where S3 is the blowup of P 2 at three points. The first part of this Theorem has been conjectured in [6] for any smooth Fano variety X, generalizing a conjecture by S. Mukai. Observe that after Mori theory, we always have ιX ≤ n+1 [13]; moreover ιX = n+1 if and only if X ∼ = Pn [9]. For a smooth Fano variety X, (i) is known in the cases ιX ≥ 1n + 1 [20, 15], 2 n ≤ 4 [6], n = 5 [2], and, provided that X admits an unsplit covering family of rational curves, ιX ≥ 1n + 1 [2]. For X toric, (i) was already known in the cases 3 n ≤ 7 or ιX ≥ 1 n + 1 [6]. 3 The inequality in (ii) was conjectured by V. V. Batyrev (see [11, page 337]) and was already known to hold up to dimension 5 (for n ≤ 4 thanks to the classifications [3, 19, 5, 17], and for n = 5 it is [8, Theorem 4.2]). Recently B. Nill [14] has shown (ii) for some classes of toric Fano varieties (see below). Observe that the bound in (ii) does not hold for non toric Fano varieties, already in dimension two. It is remarkable that in the non toric case, there is no known bound for the Picard number of a smooth Fano variety in terms of its dimension (at least to our knowledge). If S is a surface obtained blowingup P2 at eight points, for any even n the variety S n 2 has Picard number 9n, and one 2 could conjecture this is the maximum for any dimension n (see [10, page 122]). For a smooth toric Fano X, V. E. Voskresenskiĭ and A. Klyachko have shown that ρX ≤ n2 − n + 1 [18, Theorem 1]; O. Debarre has improved this bound in
NOTES ON THE ROOTS OF EHRHART POLYNOMIALS
, 2006
"... We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n 2, where n is the dimension. This improves on the previously best known bound n an ..."
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We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n 2, where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun [8] where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n 2. For the class of 0symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes. We further give a characterisation of the roots of the Ehrhart polyomials in the 3dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part1/2. These polytopes belong to the class of reflexive polytopes.
Displacing Lagrangian toric fibers via probes
"... Abstract. This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the poly ..."
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Abstract. This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya–Oh–Ohta–Ono, we show that every rational polytope has a central point that is not displaceable by probes. In the monotone (or more generally, the reflexive) case, this central point is its unique interior integral point. In the monotone case, every other point is displaceable by probes if and only if the polytope satisfies the star Ewald condition. (This is a strong version of the Ewald conjecture concerning the integral symmetric points in the polytope.) Further, in dimensions up to and including three every monotone polytope is star Ewald. These results are closely related to the Fukaya–Oh–Ohta– Ono calculations of the Floer homology of the Lagrangian fibers of a toric symplectic manifold, and have applications to questions introduced by Entov–Polterovich about the displaceability of these fibers. 1.
Classification of toric Fano 5folds
"... Abstract. We obtain 866 isomorphism classes of fivedimensional nonsingular toric Fano varieties using a computer program and the database of fourdimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes having small integral distance from any vertex. ..."
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Abstract. We obtain 866 isomorphism classes of fivedimensional nonsingular toric Fano varieties using a computer program and the database of fourdimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes having small integral distance from any vertex.
THE REFLEXIVE DIMENSION OF A LATTICE POLYTOPE
, 2004
"... The reflexive dimension refldim(P) of a lattice polytope P is the minimal d so that P is the face of some ddimensional reflexive polytope. We show that refldim(P) is finite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k. ..."
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Cited by 13 (1 self)
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The reflexive dimension refldim(P) of a lattice polytope P is the minimal d so that P is the face of some ddimensional reflexive polytope. We show that refldim(P) is finite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k.
On the combinatorial classification of toric log Del Pezzo surfaces
 LMS Journal of Computation and Mathematics
"... Abstract. Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in terms of the index ℓ. Techniques for clas ..."
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Cited by 11 (3 self)
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Abstract. Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in terms of the index ℓ. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all ℓ≤16 is obtained. 1.
Examples of nonsymmetric KählerEinstein toric Fano manifolds
"... Abstract. In this note we report on examples of 7 and 8dimensional toric Fano manifolds that are not symmetric and still admit a KählerEinstein metric. This answers a question first posed by V.V. Batyrev and E. Selivanova. The examples were found in the classification of ≤ 8dimensional toric Fan ..."
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Abstract. In this note we report on examples of 7 and 8dimensional toric Fano manifolds that are not symmetric and still admit a KählerEinstein metric. This answers a question first posed by V.V. Batyrev and E. Selivanova. The examples were found in the classification of ≤ 8dimensional toric Fano manifolds obtained by M. Øbro. We also discuss related open questions and conjectures. 1.
Classification of pseudosymmetric simplicial reflexive polytopes
, 2005
"... Gorenstein toric Fano varieties correspond to so called reflexive polytopes. If such a polytope contains a centrally symmetric pair of facets, we call the polytope, respectively the toric variety, pseudosymmetric. Here we present a complete classification of pseudosymmetric simplicial reflexive po ..."
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Cited by 10 (5 self)
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Gorenstein toric Fano varieties correspond to so called reflexive polytopes. If such a polytope contains a centrally symmetric pair of facets, we call the polytope, respectively the toric variety, pseudosymmetric. Here we present a complete classification of pseudosymmetric simplicial reflexive polytopes. This is a generalization of a result of Ewald on pseudosymmetric nonsingular toric Fano varieties and recent work of Wirth. As applications we determine the maximal number of vertices, facets and lattice points, and show that the vertices can be chosen to have coordinates −1,0, 1.