We classify terminal simplicial reflexive d-polytopes with 3d − 1 vertices. They turn out to be smooth Fano d-polytopes. When d is even there is 1 such polytope up to isomorphism, while there are 2 when d is uneven.
In dimension d, Q-factorial Gorenstein toric Fano varieties with Picard number ρ X correspond to simplicial reflexive polytopes with ρ X +d vertices. Casagrande showed that any d-dimensional simplicial reflexive polytope has at most 3d vertices, if d is even, respectively, 3d − 1, if d is odd. Moreover, for d even there is up to unimodular equivalence only one such polytope with 3d vertices, corresponding to (S 3 ) d/2 with Picard number 2d, where S 3 is the blow-up of P 2 at three non collinear points. In this paper we completely classify all d-dimensional simplicial reflexive polytopes having 3d − 1 vertices, corresponding to d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number 2d − 1. For d even, there exist three such varieties, with two being singular, while for d > 1 odd there exist precisely two, both being nonsingular toric fiber bundles over P 1 . This generalizes recent work of the second author.
We present an algorithm that produces the classification list of smooth Fano d-polytopes 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 d-polytopes for d ≤ 7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.Recently a complete classification of smooth Fano 5-polytopes has been announced ([12]). The approach is to recover smooth Fano d-polytopes from their image under the projection along a vertex. This image is a reflexive (d − 1)-polytope (see [3]), which is a fully-dimensional lattice polytope containing the origin in the interior, such that the dual polytope is also a lattice polytope. Reflexive polytopes have been classified up to dimension 4 using the computer program PALP ([10], [11]). Using this classification and PALP the authors of [12] succeed in classifying smooth Fano 5-polytopes.In this paper we present an algorithm that classifies smooth Fano d-polytopes for any given d ≥ 1. We call this algorithm SFP (for Smooth Fano Polytopes). The input is the positive integer d, nothing else is needed. The algorithm has been implemented in C++, and used to classify smooth Fano d-polytopes for d ≤ 7. For d = 6 and d = 7 our results are new:Theorem 1.1. There are 7622 isomorphism classes of smooth Fano 6polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.The classification lists of smooth Fano d-polytopes, d ≤ 7, are available on the authors homepage: http://home.imf.au.dk/oebro A key idea in the algorithm is the notion of a special facet of a smooth Fano d-polytope (defined in section 3.1): A facet F of a smooth Fano d-polytope is called special, if the sum of the vertices of the polytope is a non-negative linear combination of vertices of F . This allows us to identify a finite subset W d of the lattice Z d , such that any smooth Fano d-polytope is isomorphic to one whose vertices are contained in W d (theorem 3.6). Thus the problem of classifying smooth Fano d-polytopes is reduced to the problem of considering certain subsets of W d . We then define a total order on finite subsets of Z d and use this to define a total order on the set of smooth Fano d-polytopes, which respects isomorphism (section 4). The SFP-algorithm (described in section 5) goes through certain finite subsets of W d in increasing order, and outputs smooth Fano d-polytopes in increasing order, such that any smooth Fano d-polytope is isomorphic to exactly one in the output list. As a consequence of the total order on smooth Fano d-polytopes, the algorithm needs not consult the previous output to check for isomorphism to decide whether or not to output a constructed polytope. Smooth Fano polytopesWe fix a notation and prove some simple facts about smooth Fano polytopes.
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