# A bound for the splitting of smooth Fano polytopes with many vertices

@article{Assarf2014ABF, title={A bound for the splitting of smooth Fano polytopes with many vertices}, author={Benjamin Assarf and Benjamin Nill}, journal={Journal of Algebraic Combinatorics}, year={2014}, volume={43}, pages={153-172} }

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior and is such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least $$3d-2$$3d-2 vertices are completely known. The main…

## One Citation

Webs of stars or how to triangulate free sums of point configurations

- Computer Science, MathematicsJ. Comb. Theory, Ser. A
- 2018

## References

SHOWING 1-10 OF 17 REFERENCES

Q-factorial Gorenstein toric Fano varieties with large Picard number

- Mathematics
- 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…

On the Picard number of divisors in Fano manifolds

- Mathematics
- 2009

Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image H of N_1(D) in N_1(X) under the natural push-forward of 1-cycles. We show that the…

Smooth Fano Polytopes with Many Vertices

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2014

It turns out that all of the d-dimensional simplicial, terminal, and reflexive polytopes with at least 3d-2-2 vertices are smooth Fano poly topes.

Toric Fano varieties and birational morphisms

- Mathematics
- 2001

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have…

Toward the classification of higher-dimensional toric Fano varieties

- Mathematics
- 1999

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By…

An algorithm for the classification of smooth Fano polytopes

- Mathematics
- 2007

We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm…

On the classification of toric Fano 4-folds

- Mathematics
- 1998

AbstractThe biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝd, the so-called Fano polyhedra, up to an…

Numerical invariants of Fano 4-folds

- Mathematics
- 2012

Let X be a Fano 4-fold. For any prime divisor D in X, consider where i* is the push-forward of one-cycles, and let cD be the codimension of in . We define an integral invariant cX of X as the maximal…

Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

- Mathematics
- 1993

We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials…

Classification of toric Fano 5-folds

- Mathematics, Physics
- 2007

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…