Smooth Fano polytopes can not be inductively constructed

Abstract

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.

On smooth Gorenstein polytopes

“…By now, toric Fano manifolds are known up to dimension 9 by Øbro's algorithm [Øb07,LP] extending previous classifications [WW82, Bat81, Bat99, Sat06, KN09]. As an application of our algorithmic results (Table 1) combined with the previous proposition we can determine all non-isomorphic toric Fano manifolds with…”

Applications of the moduli continuity method to log K‐stable pairs

“…Remark Notice that there are known examples of smooth toric Fanos having the same degree as the projective space starting from dimension 5 (the unique projective toric smooth Fano 5‐fold with whose toric polytope has 8 vertices, 18 facets and , [9, 46]). Thus, just having the same volume is not sufficient to determine that and we really use that the index is preserved.…”

Equivalence classes for smooth Fano polytopes

“…Moreover, each of the double circled numbers corresponds to a smooth Fano 5-polytope P such that for any Q ∈ F (5) \ {P }, Q is not I-equivalent to P (i.e., I-isolated, see Section 4). Note that Øbro's example [11] corresponds to the double circled number 164. In addition, if two circled numbers are connected by a line, then those are F-equivalent.…”

“…Let F (n) be the set of all unimodular equivalence classes for smooth Fano npolytopes. The main concern of this paper is some equivalence classes for F (n) with respet to the following two equivalence relations: Definition 1.1 ([12, Definition 1.1, 6.1], [11,Definition 1.1]). We say that two smooth Fano n-polytopes P and Q are F-equivalent if there exists a sequence P 0 , P 1 , .…”

“…This is true when n ≤ 4. However, Øbro [11] has given a counterexample of this conjecture. His counterexample is a smooth Fano 5-polytope P with 8 vertices which is neither pseudo-symmetric nor I-equivalent to any other smooth Fano 5-polytope, i.e., for any Q ∈ F (5) \ {P }, Q is never I-equivalent to P .…”