Dual del Pezzo 4Polytope
Benjamin Assarf, Michael JoswigMedia
Description
The del Pezzo polytope of dimension $d$ is the convex hull of the $2d+2$ points $\{\pm e_1,\pm e_2, \dots, \pm e_d, \pm \mathbf{1}\}$. The DelPezzo polytopes are lattice polytopes which are terminal and reflexive. If $d$ is even the del Pezzo $d$polytope is a smooth Fano polytope.
A $\textit{lattice polytope}$ $P$ is a convex polytope whose vertices lie in the lattice $\mathbb{Z}^d$ contained in the vector space $\mathbb{R}^d$. Fixing a basis of $\mathbb{Z}^d$ describes an isomorphism to $\mathbb{Z}^d$. A $d$dimensional lattice polytope $P\subset\mathbb{R}^d$ is called $\textit{reflexive}$ if it contains the origin $0$ as an interior point and its polar polytope is a lattice polytope. A lattice polytope $P$ is $\textit{terminal}$ if $0$ and the vertices are the only lattice points in $P\cap \mathbb{Z}^d$. It is $\textit{simplicial}$ if each face is a simplex. We say that $P$ is a $\textit{smooth Fano polytope}$ if $P\subseteq \mathbb{R}^d$ is simplicial with $0$ in the interior and the vertices of each facet form a lattice basis of $\mathbb{Z}^d$. The fan where every cone is the non negative linear span over a face is called the $\textit{face fan}$. It is dual to the $\textit{normal fan}$, which is the collection of all normal cones.
The interest in classifications of smooth Fano polytopes has its origins in applications of algebraic geometry to mathematical physics. For instance, Batyrev [1] uses reflexive polytopes to construct pairs of mirror symmetric CalabiYau manifolds; see also [2].
In algebraic geometry, reflexive polytopes correspond to $\textit{Gorenstein toric Fano varieties}$. The toric variety $X_P$ of a polytope $P$ is determined by the face fan of $P$, that is, the fan spanned by all faces of $P$; see Ewald [3] or Cox, Little, and Schenck [4]. The toric variety $X_P$ is $\mathbb{Q}$$\textit{factorial}$ (some multiple of a Weil divisor is Cartier) if and only if the polytope $P$ is simplicial. In this case the $\textit{Picard number}$ of $X$ equals $nd$, where $n$ is the number of vertices of $P$. The polytope $P$ is smooth if and only if the variety $X_P$ is a manifold (that is, it has no singularities).
In 2006 Casagrande [5] proved that the number of vertices of $d$di\men\sional simplicial, terminal, and reflexive lattice polytopes does not exceed $3d$. Those polytopes are classified if the number of vertices is greater or equal $3d2$ (see [5] [6] [7]).
If you have two smooth Fano polytopes $P$ and $Q$, then the free sum $P\oplus Q$ is also smooth Fano. The same holds for a bipyramid over a smooth Fano polytope. A natural question to ask is how many smooth Fano polytopes are a free sum or a bipyramid over lower dimensional ones.
$\textbf{Conjecture:}$ Let $P$ be a $d$dimensional smooth Fano polytope with $n$ vertices such that $n\ge 3dk$ for $k\le\tfrac{d}{3}$. If $d{+}k$ is even then $P$ is lattice equivalent to $Q\oplus P_6^{\oplus (\frac{d3k}{2})}$ where $Q$ is a $3k$dimensional smooth Fano polytope with $n3d+9k\ge 8k$ vertices. If $d{+}k$ is odd then $P$ is lattice equivalent to $Q\oplus P_6^{\oplus (\frac{d3k1}{2})}$ where $Q$ is a $(3k{+}1)$dimensional smooth Fano polytope with $n3d+9k3\ge 8k3$ vertices.
A recent result [8] supports this conjecture.
One example of a polytope which is not a free sum or a bipyramid over some lowerdimensional smooth Fano polytope is the del Pezzo $4$polytope, which has $10 = 3\cdot 42$ vertices and $30$ facets. A Schlegel diagram of the dual del Pezzo $4$polytope is visualized in [Fig 1]. One can download the $4$dimensional del Pezzo polytope as a $\texttt{polymake}$ file in [data]. For $\texttt{polymake}$ check: www.polymake.org
References

Victor V. Batyrev.
Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties.
J. Alg. Geom, pages 493–535, 1994.

Victor V Batyrev and Lev A Borisov.
Mirror duality and stringtheoretic Hodge numbers.
Inventiones mathematicae, 126(1):183–203, 1996.

Günter Ewald.
Combinatorial convexity and algebraic geometry.
Springer, 1996.

David A. Cox, John B. Little, and Henry K. Schenck.
Toric varieties.
American Mathematical Society, 2011.

Cinzia Casagrande.
The number of vertices of a Fano polytope.
Annales de l'Institut Fourier, 56(1):121–130, 2006.
URL: https://aif.centremersenne.org/item/AIF_2006__56_1_121_0, doi:10.5802/aif.2175. 
Mikkel Øbro.
Classification of terminal simplicial reflexive dpolytopes with 3d − 1 vertices.
manuscripta mathematica, 125(1):69–79, January 2008.
doi:10.1007/s002290070133z. 
Benjamin Assarf, Michael Joswig, and Andreas Paffenholz.
Smooth Fano Polytopes with Many Vertices.
Discrete Computational Geometry, Springer US, 2014(52):153–194, 2014.
doi:10.1007/s0045401496074. 
Benjamin Assarf and Benjamin Nill.
A bound for the splitting of smooth Fano polytopes with many vertices.
Journal of Algebraic Combinatorics, 43(1):153–172, 2016.
doi:10.1007/s1080101506301.