## Toric Geometry I – Definitions

Ok, so I’m going to mostly be posting on toric geometry for the near future, and in particular, working out what we need to do some mirror symmetry with it. I’ll be following various things by David Cox, so check out his webpage for more info. Anyway, this post is mostly going to be background on fans, polytopes, and cones so that we can do toric geometry properly next time. In fact, varieties will not be mentioned.

Before getting to the math, a personal note. The stuff in this post is rather near to my heart, as Coxeter’s Regular Polytopes was my first real math book, and my interest in mathematics really traces back to Flatland and my high school geometry class. So all in all, I’m rather glad that I’ve found a way to take classical geometry and use it for algebraic geometry.

Anyway, a lattice is a finitely generated free abelian group. Take a lattice $M$ and its dual lattice $N=\hom(M,\mathbb{Z})$. There is a pairing $M\times N\to \mathbb{Z}$ given by $(m,n)=n(m)$. We’ll denote by $M_{\mathbb{R}}$ and $N_{\mathbb{R}}$ the vector spaces obtained by tensoring with the real numbers.

So now, a rational polyhedral cone $\sigma\subseteq N_{\mathbb{R}}$ is just the linear combinations of a finite collection of elements of $N$ where all the coefficients are positive. That is, $\sigma=\{\lambda_1u_1+\ldots+\lambda_su_s|\lambda_1,\ldots,\lambda_s\geq 0\}$ and $u_1,\ldots,u_s\in N$ are fixed. We say that $\sigma$ is strongly convex if $\sigma\cap (-\sigma)=\{0\}$. What this means is that it doesn’t contain any complete lines. Without it, a rational polyhedral cone is still convex, but it could be as big as a half space. Strong convexity rules that out. We say $\dim\sigma=\dim \mathrm{Span}(\sigma)$.

Now, for any cone $\sigma$ (we’ll often just forget the rational polyhedral part, because those are the only cones we really care about), a face is the intersection with the set of points $\ell=0$, where $\ell$ is a linear form which is nonnegative on $\sigma$. Denote by $\sigma(r)$ the set of faces of dimension $r$. Any face will, in fact, be a subcone. Call an element $\rho\in \sigma(1)$ an edge, and the primitive element $n_\rho$ corresponding to the edge is the generator of $\rho\cap N$. It’s fairly straightforward (even by just drawing a picture, though that’s not rigorous) to see that the $n_\rho$ generate $\sigma$, that is, they’re the elements appearing in the definition above, the $u_i$‘s. Now, that’s dimension one. In codimension one, we have facets. Even better, if $\sigma$ has the same dimension as $N_{\mathbb{R}}$, then there’s a unique element of $M_{\mathbb{R}}$ defined by being normal (If the pairing above is denoted $\langle m,n\rangle$, then $\langle m,n\rangle=0$ for all $n$ in the facet) and having minimal length.

Now, given a strongly convex cone, we define the dual cone $\sigma^\vee=\{m\in M_{\mathbb{R}}| \langle m,u\rangle\geq 0$ for all $u\in \sigma\}$.

Now we get to the first properly important bit: fans. A fan is a finite collection $\Sigma$ of cones in $N_{\mathbb{R}}$ such that they’re all strongly convex, if $\sigma\in \Sigma$ and $\tau$ is a face of $\sigma$, then $\tau\in\Sigma$ as well, and the intersection of pairs of cones in $\Sigma$ is a face of each. These conditions will let us use fans to define toric varieties next time. We’ll define a bunch of properties of cones that will comes up:

A cone is smooth if it is generated by a subset of a basis for $N$. It is simplicial if it is generated by a subset of a basis of $N_{\mathbb{R}}$. We’ll come back to all of these later.

Now, we move from cones and fans, and start discussing polytopes. A lattice polytope $\Delta$ is just the convex hull in $M_{\mathbb{R}}$ of a finite subset of $M$. Now, for any facet, we have an inward normal vector $n_F\in N$ of minimal length and integer $a_F$, so that $\Delta$ is given by the inequalities $\langle m,n_F\rangle\geq -a_F$ for all $F$ facets.

Given a lattice polytope, we can define a fan as follows: take any face $f$, and set $\sigma_f$ to be the cone generated by $n_F$ for the facets containing $f$. Then $\Sigma_\Delta=\{\sigma_f|f$ is a face of $\Delta\}$ is a fan.

Additionally, given a polytope $\Delta\subset N_{\mathbb{R}}$, there exists a polytope $\Delta^\circ\subset M_{\mathbb{R}}$ which is called the dual polytope, given by $\Delta^\circ=\{m\in M_{\mathbb{R}}|\langle m,u\rangle\geq -1$ for all $u\in\Delta\}$.

Now, an $n$-dimensional polytope in $M_{\mathbb{R}}$ is called reflexive if all the facets $F$ of $\Delta$ are supported by an affine hyperplane, that is, they’re the intersection of the polytope with $\{m\in M_{\mathbb{R}}|\langle m,n_F\rangle=-1\}$ for some $n_F\in N$ and if the only interior lattice point is 0. It’s a theorem of Batyrev that $\Delta$ is reflexive if and only if $\Delta^\circ$ is.

So now, lastly, we devote a moment to thinking about the dual polytope geometrically. Take the cube with vertices $\pm e_1\pm e_2\pm e_3$ in three space. Then, the inward normals are $\pm e_1,\pm e_2,\pm e_3$, and so it has dual with those vertices, and that’s the octahedron. Now, to get the fan corresponding to any polytope $\Delta$, you look at its dual $\Delta^\circ$, and then take the cones over the faces and put them all together. They’ll form a fan, and this is the fan you’d get from the definition above. So the fan corresponding to the cube is just the octants, and the fan corresponding to the octahedron is a collection of infinite pyramids. And finally, we call the fans arising from dual polytopes dual. So we can work on the level of fans or polytopes at our leisure, and will use them both.

Apologies on the lack of pictures, I’ve failed at creating any to put here.