# Wednesday, August 30

:::{.remark}
Given a polytope $P$, any integral affine transform, i.e. an element of $\GL_n(\ZZ) \semidirect \ZZ^n$, produces an isomorphic toric variety.
One can form a fan by taking normal vectors to edges of $P$; from a fan one can define an abstract toric variety (independent of projective embeddings).
Let $\Sigma_P$ be the normal fan associated to $P$.
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:::{.definition title="Fan"}
A fan $\Sigma \subseteq N_\RR$ is a collection of strongly convex rational polyhedral cones $\ts{\sigma}$ satisfying

- $\sigma\in \Sigma \implies$ all faces of $\sigma$ are in $\Sigma$.
- \( \sigma, \sigma' \in \Sigma \implies \sigma \intersect \sigma' \) is a face of both \( \sigma \) and \( \sigma' \).

:::

:::{.remark}
One can construct for every face $Q$ of $P$, a cone $\sigma_Q$ of $\Sigma_P$ and realize $\Sigma_P = \ts{\sigma_Q}$.
Codimension 0 faces correspond to points in $\Sigma_P$, and codimension 1 faces $Q$ correspond to affine hyperplanes.

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\end{tikzpicture}

Sign choice: given a hyperplane, choose a normal vector to get a linear form. Choose the sign so that this is positive.
For higher codimension faces $Q$: look at all codimension 1 faces containing $Q$, take the associated ray for each such face, and let $\sigma_Q$ be the cone spanned by all such rays.
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:::{.remark}
For a vertex $v\in P$, one can describe $\sigma_v$ in a different way: take a neighborhood of $v$, intersect with the interior of $p$, regard that as a cone, and take the dual cone.

\begin{tikzpicture}
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\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2023/Fall/Toric/sections/figures}{2023-08-30_14-32.pdf_tex} };
\end{tikzpicture}

Note that $\spec \CC[\sigma_v\dual \intersect M]$ is an affine chart for the projective variety defined by $P$.
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:::{.remark}
Note that if $P$ is a polytope, both $P$ and $kP$ have the same normal fan for any $k \in \ZZ_{> 0}$.
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