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- VICTOR BATYREV, Benjamin Nill
- 2007

Let âˆ† be an n-dimensional lattice polytope. The smallest non-negative integer i such that kâˆ† contains no interior lattice points for 1 â‰¤ k â‰¤ nâˆ’ i we call the degree of âˆ†. We consider lattice polytopes of fixed degree d and arbitrary dimension n. Our main result is a complete classification of n-dimensional lattice polytopes of degree d = 1. This is aâ€¦ (More)

- VICTOR BATYREV, Benjamin Nill
- 2007

The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index râ€¦ (More)

- Benjamin Nill
- 2008

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 generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants andâ€¦ (More)

- Benjamin Nill
- Eur. J. Comb.
- 2008

The hâˆ—-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with hâˆ—-polynomial of degree d and with linear coefficient hâˆ— 1 . We show that P has to be a lattice pyramid over a lower-dimensional lattice polytope, if the dimension of P is greater or equal to hâˆ— 1 (2d + 1) + 4d âˆ’ 1.â€¦ (More)

- Benjamin Nill
- 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, pseudo-symmetric. Here we present a complete classification of pseudo-symmetric simplicial reflexive polytopes. This is a generalization of a result of Ewald onâ€¦ (More)

- Alexander M. Kasprzyk, Benjamin Nill
- Electr. J. Comb.
- 2012

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of l-reflexiveâ€¦ (More)

- Benjamin Nill
- 2004

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 theâ€¦ (More)

- Benjamin Nill
- Discrete & Computational Geometry
- 2007

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d â‰¥ 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. Inâ€¦ (More)

- Christian Haase, Benjamin Nill, Andreas Paffenholz, Francisco Santos
- Electr. J. Comb.
- 2008

Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P .

- Christian Haase, Benjamin Nill
- J. Comb. Theory, Ser. A
- 2008

Lattices generated by lattice points in skeletons of reflexive polytopes are essential in determining the fundamental group and integral cohomology of Calabi-Yau hypersurfaces. Here we prove that the lattice generated by all lattice points in a reflexive polytope is already generated by lattice points in codimension two faces. This answers a question of J.â€¦ (More)