The toric variety of a polytope and the normal fan

(Faces)

Let $P \subseteq M_\RR$ be a lattice polytope. A face is a slice by a supporting affine hyperplane, \[ F = P \intersect \ts{m \in M_\RR \st \inp{m}{u} = r}, \qquad \inp{m}{u} \geq r \text{ on } P , \] for some $u \in N$ and $r \in \RR$. The facets are the faces of codimension one, and $P$ has the facet presentation \[ P = \ts{ m \in M_\RR \st \inp{m}{u_F} \geq -a_F \text{ for every facet } F } , \] where $u_F \in N$ is the primitive inward normal to $F$.

(The normal fan)

For a face $F \leq P$ set \[ \sigma_F = \Cone\big( u_F' \st F \leq F',\ F' \text{ a facet} \big) \subseteq N_\RR . \] These cones form the normal fan $\Sigma_P$, and $X_P \da X_{\Sigma_P}$. Vertices of $P$ give the maximal cones, facets of $P$ give the rays, and the whole poset is reversed. The vertex $m_i$ corresponds to the maximal cone $\sigma_i = \dualof{\Cone(P \cap M - m_i)}$.

(Combinatorial equivalence)

Polytopes $P_1$ and $P_2$ are combinatorially equivalent if there is a bijection between their faces that preserves inclusions, intersections and dimensions of faces.

(A hexagon)

For the hexagon $P$ with vertices $v_1, \ldots, v_6$, the cone of directions out of each vertex $v_i$ is dual to the maximal cone $\sigma_i$ of the normal fan $\Sigma_P$, which has six maximal cones.

A lattice hexagon P with the edge directions at each vertex, and its normal fan \Sigma_P. — A lattice hexagon $P$ with the edge directions at each vertex, and its normal fan $\Sigma_P$.

(Descriptions of $X_P$)

Let $P$ be a full-dimensional lattice polytope.

$X_P$ is the toric variety of the normal fan $\Sigma_P$.
If $0$ lies in the interior of $P$, then $\Sigma_P$ is the fan of cones over the proper faces of the polar dual $P^\circ$.
For a vertex $m$ of $P$, the maximal cone $\sigma_m$ of $\Sigma_P$ has dual cone $\dualof{\sigma_m}=\Cone(P \intersect M - m)$, the cone of directions out of $m$, and $X_P = \Union_{m \text{ a vertex}} \Spec k[\Cone(P \intersect M - m) \intersect M]$.

$X_P$ carries the ample divisor $D_P$ of D-TORQD, so every $X_P$ is projective. Conversely, a complete fan need not be the normal fan of a polytope, and a complete toric variety need not be projective.

What is the toric variety attached to a lattice polytope?

What is the normal fan of a polytope, and how does it relate to cones over faces?

Dependencies

D-Q7Q2NFrom a fan to a toric varietyuses

Backlinks

D-FULNORMPOLYNormal polytopes and very ample dilationsuses
D-FULSIMPLESimple and simplicial polytopes and simplicial normal fansuses
T-FULDEGThe degree of a projective toric variety is the normalised volume of its polytopeuses
D-TORMOMENTThe moment map and the polytope as an orbit spaceuses
D-TORQDThe polytope of a divisor, and its lattice points as sectionsuses

What links to this

Polytopes and divisors