Benjamin Nill

Gorenstein toric Fano varieties

Abstract.We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalizations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.

Complete toric varieties with reductive automorphism group

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.

Cayley decompositions of lattice polytopes and upper bounds for h*-polynomials

Abstract We give an effective upper bound on the h*-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. Polytopes with nontrivial Cayley decompositions correspond to projectivized sums of toric line bundles, and our approach is partially inspired by classification results of Fujita and others in algebraic geometry. In an appendix, we interpret our Cayley decomposition theorem in terms of adjunction theory for toric varieties.

Classification of toric Fano 5-folds

We obtain 866 isomorphism classes of five-dimensional nonsingular toric Fano varieties using a computer program and the database of four-dimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes having small integral distance from any vertex.

Volume and Lattice Points of Reflexive Simplices

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected to contain the largest number of lattice points even among all lattice polytopes with only one interior lattice point. Translated in algebro-geometric language, our main theorem yields a sharp upper bound on the anticanonical degree of d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, e.g., of weighted projective spaces with Gorenstein singularities.

Projecting Lattice Polytopes Without Interior Lattice Points

We show that up to unimodular equivalence in each dimension there are only finitely many lattice polytopes without interior lattice points that do not admit a lattice projection onto a lower-dimensional lattice polytope without interior lattice points. This was conjectured by Treutlein [Treutlein, J. 2008. 3-Dimensional lattice polytopes without interior lattice points. September 10, http://arXiv.org/abs/0809.1787.] As an immediate corollary, we get a short proof of a recent result of Averkov, Wagner, and Weismantel [Averkov, G., C. Wagner, R. Weismantel. 2010. Maximal lattice-free polyhedra: Finiteness and an explicit description in dimension three. Math. Oper. Res. Forthcoming.], namely, the finiteness of the number of maximal lattice polytopes without interior lattice points. Moreover, we show that, in dimension four and higher, some of these finitely many polytopes are not maximal as convex bodies without interior lattice points.

Polyhedral adjunction theory

In this paper we offer a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we explore two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We prove a structure theorem for lattice polytopes P with large Q-codegree. For this, we define the adjoint polytope P-(s) as the set of those points in P whose lattice distance to every facet of P is at least s. It follows from our main result that if P-(s) is empty for some s < 2/(dim P + 2), then the lattice polytope P has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

On the combinatorial classification of toric log del Pezzo surfaces

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<17 is obtained.

Finitely many smooth d-polytopes with N lattice points

We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙn that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.

Reconstruction and virtual model of the Schickard calculator

Abstract Exhibition of objects such as paintings or historical artefacts often involves a common problem: the objects presented are unique, delicate and, therefore, very valuable. On the other hand, these objects should be made accessible to scholars and educators. We present an application of modern 3D computer graphics in the field of reconstructing ancient scientific instruments. The first-four-species calculator of Wilhelm Schickard is made accessible to the public in the World Wide Web using Java 3D.