Toric

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motivation

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toric varieties

Ideas: - A variety $X$ containing a dense open algebraic torus $T$ such that $T\curvearrowright T$ extends to an action $T\curvearrowright X$. - A normal variety $X$ containing a torus $T$ such that $X/T$ has finitely many orbits. - A variety described as the closure of the image of a monomial map $f: (k^{\times})^m \to {\mathbf{A}}^n_{/ {k}} $ where $\mathbf{x} \mapsto {\left[ {\mathbf{x}^{\mathbf{a}_1}, \cdots, \mathbf{x}^{\mathbf{a}_n}} \right]}$.

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Notation

$M$ is a lattice over a field $k$ with a valuation.
$N = M {}^{ \vee }\coloneqq\mathop{\mathrm{Hom}}_{ {}_{{\mathbf{Z}}}{\mathsf{Mod}} }(N, {\mathbf{Z}})$ is the dual lattice.
$T_N = N \otimes_{\mathbf{Z}}k^{\times}= \mathop{\mathrm{Hom}}(M, k^{\times}) = (k^{\times})^n$ is the torus.
$N_{\mathbf{R}}\coloneqq N\otimes_{\mathbf{Z}}{\mathbf{R}}\cong {\mathbf{R}}^n$ is the real form of $N$.
$\Sigma$ is a rational fan in $N_{\mathbf{R}}$
$X_\Sigma$ is the associated toric variety completing $T_N$.
$\rho_i$ are the rays generating $\Sigma$
$v_i$ are the first lattice points along the rays $\rho_i$
$V$ is the matrix whose colums are the $v_i$.
$\sigma$ are cones in $\Sigma$.
$S$ is the Cox ring.
$B$ is the irrelevant ideal.

Fans

Toric varieties are entirely determined by their associated fan: a collection of cones closed under taking intersections and faces. In special cases, this is further determined by a polytope.

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A fan $\mathcal{G}$ is a family of nonempty closed polyhedral (convex) cones in $V$ such that
- Every face of a cone in $\mathcal{G}$ is in $\mathcal{G}$, and
- The intersection of any two cones in $\mathcal{G}$ is a face of both.

Properties:

A fan $\mathcal{G}$ is complete if the union of all its cones is $V$,
${\mathcal{G}}$ is essential (or pointed) if the intersection of all non-empty cones of $\mathcal{G}$ is the origin
${\mathcal{G}}$ is simplicial if every cone is simplicial, that is, spanned by linearly independent vectors.
A 1-dimensional cone is called a ray.
A ray is extremal if it is a face of some cone.
The set of $k$-dimensional cones of $\mathcal{G}$ is denoted by $\mathcal{G}^{(k)}$ and two cones in $\mathcal{G}^{(k)}$ are adjacent if they have a common face in $\mathcal{G}^{(k-1)}$.
A fan $\mathcal{G}$ coarsens a fan $\mathcal{G}^{\prime}$ if every cone of $\mathcal{G}$ is the union of cones of $\mathcal{G}^{\prime}$ and $\bigcup_{C \in \mathcal{G}} C=\bigcup_{C \in \mathcal{G}^{\prime}} C$.