Examples of non-symmetric K\"ahler-Einstein toric Fano manifolds (2009)
Nill, Benjamin, Paffenholz, Andreas
In this note we report on examples of 7- and 8-dimensional toric Fano manifolds that are not symmetric and still admit a Kaehler-Einstein metric. This answers a question first posed by V.V. Batyrev...
On the combinatorial classification of toric log del Pezzo surfaces (2008)
Kasprzyk, Alexander M., Kreuzer, Maximilian, Nill, Benjamin
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...
Q-factorial Gorenstein toric Fano varieties with large Picard number (2008)
Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial...
Cayley decompositions of lattice polytopes and upper bounds for h^*-polynomials (2008)
Haase, Christian, Nill, Benjamin, Payne, Sam
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...
Reflexive Polytopes- Combinatorics and Convex Geometry (2008)
· What is a reflexive polytope? · How many are there? · What special properties do they have? · What classification results exist? · Bounds on invariants as the vertices or the volume? · What...
Lattice points in Minkowski sums (2007)
Haase, Christian, Nill, Benjamin, Paffenholz, Andreas, Santos, Francisco
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...
On permutation polytopes (2007)
Baumeister, Barbara, Haase, Christian, Nill, Benjamin, Paffenholz, Andreas
A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely...
A boundedness result for toric log Del Pezzo surfaces (2007)
Dais, Dimitrios I., Nill, Benjamin
In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\ell$. This upper bound...
Lattice polytopes having h^*-polynomials with given degree and linear coefficient (2007)
The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient...
Combinatorial aspects of mirror symmetry (2007)
Batyrev, Victor, Nill, Benjamin
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...
Classification of toric Fano 5-folds (2007)
Kreuzer, Maximilian, Nill, Benjamin
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...
Multiples of lattice polytopes without interior lattice points (2006)
Batyrev, Victor, Nill, Benjamin
Let $\Delta$ be an $n$-dimensional lattice polytope. The smallest non-negative integer $i$ such that $k \Delta$ contains no interior lattice points for $1 \leq k \leq n - i$ we call the degree of...
Classification of pseudo-symmetric simplicial reflexive polytopes (2005)
A reflexive polytope, respectively its associated Gorenstein toric Fano variety, is called pseudo-symmetric, if the polytope has a centrally symmetric pair of facets. Here we present a complete...
Gorenstein toric Fano varieties (2005)
In this thesis we concern ourselves with Gorenstein toric Fano varieties, that is, with complete normal toric varieties whose anticanonical divisor is an ample Cartier divisor. These...
Lattices generated by skeletons of reflexive polytopes (2005)
Haase, Christian, Nill, Benjamin
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...
Gorenstein toric Fano varieties [Elektronische Ressource] / (2005)
Tübingen, Univ., Diss., 2005.
Volume and lattice points of reflexive simplices (2004)
We prove sharp upper bounds on the volume and the number of lattice points on edges of higher-dimensional reflexive simplices. These convex-geometric results are derived from new number-theoretic...
Complete toric varieties with reductive automorphism group (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...
Gorenstein toric Fano varieties (2004)
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...