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Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course, combinatorics. In this article, we review several classes of polytopes that do have unimodular triangulations and constructions that preserve their existence. We include, in particular, the first effective proof of the classical result by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation. Our proof yields an explicit (although doubly exponential) bound for the dilation factor.

In this note we prove Fujita's spectrum conjecture for polarized varieties in the case of ℚ-Gorenstein projective toric varieties of index r. The theorem follows from a combinatorial result using the connection between lattice polytopes and polarized projective toric varieties. By this correspondence the spectral value of the polarized toric variety equals the ℚ-codegree of the polytope. Now the main theorem of the paper shows that the spectrum of the ℚ-codegree is finite above any positive threshold in the class of lattice polytopes with α-canonical normal fan for any fixed α > 0.

In this note we prove Fujita’s spectrum conjecture for polarized varieties in the case of Q\\mathbb {Q}-Gorenstein projective toric varieties of index rr. The theorem follows from a combinatorial result using the connection between lattice polytopes and polarized projective toric varieties. By this correspondence the spectral value of the polarized toric variety equals the Q\\mathbb {Q}-codegree of the polytope. Now the main theorem of the paper shows that the spectrum of the Q\\mathbb {Q}-codegree is finite above any positive threshold in the class of lattice polytopes with α\\alpha-canonical normal fan for any fixed α>0\\alpha >0.

The$d$ -dimensional simplicial, terminal, and reflexive polytopes with at least$3d-2$vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves previous results of Casagrande (2006) and Øbro (2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics. This text is an extended abstract of Assarf et al. (2012). Nous classifions les polytopes simpliciaux, terminaux et réflexifs de dimension$d$avec au moins$3d-2$sommets. En particulier, tous ces polytopes se trouvent être des polytopes de Fano lisses. Nous améliorons des résultats antérieurs de Casagrande (2006) et d’Øbro (2008). Les polytopes de Fano lisses apparaissent en géométrie algébrique et en physique mathématique. Ce texte est un résumé étendu de Assarf et al. (2012).

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets. Nous ètudions les propriètès combinatoires et gèomètriques des polytopes de permutations pour des groupes cycliques. C'est à dire, donnè un groupe cyclique de matrices de permutations, nous considèrons son enveloppe convexe. Si le gènèrateur du groupe possède un ou deux orbites il y a une dèscription simple du polytope. Par contre, le cas de trois (ou plus) orbites est beaucoup plus compliquè. Pour une classe ample d'examples nous construisons un nombre exponentiel de faces de co-dimension un.

The d -dimensional simplicial, terminal, and reflexive polytopes with at least 3 d - 2 vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves on previous results of Casagrande (Ann Inst Fourier (Grenoble) 56(1):121–130, 2006 ) and Øbro (Manuscr Math 125(1): 69–79, 2008 ). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics.

The$\\mathtt{polymake}$software system deals with convex polytopes and related objects from geometric combinatorics. This note reports on a new implementation of a subclass for lattice polytopes. The features displayed are enabled by recent changes to the$\\mathtt{polymake}$core, which will be discussed briefly.

polyDB is a database for discrete geometric objects. The database is accessible via web and an interface from the software package polymake. It contains various datasets from the area of lattice polytopes, combinatorial polytopes, matroids and tropical geometry. In this short note we introduce the structure of the database and explain its use with a computation of the free sums and certain skew bipyramids among the class of smooth Fano polytopes in dimension up to 8.

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n - 2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that \"cut across an ideal.\" Thus we arrive at a substantial generalization of Bier's construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular \"generalized Bier spheres.\" In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields \"many shellable spheres,\" most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres. [PUBLICATION ABSTRACT]

We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction does not always specialize to convex polytopes; however, in a number of cases where we can realize it, it produces interesting classes of polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple 4-polytopes, as requested by Eppstein et al. We also construct for each d 3 an infinite family of (d - 2)-simplicial 2-simple d-polytopes, thus solving a problem of Grunbaum. [PUBLICATION ABSTRACT]