# Wednesday, August 23

:::{.remark}
Today: normal affine toric varieties.
Recall $V\contains T$ is toric when $T$ is open/dense and $T\actson V$ algebraically.
Set $M\da \Hom(T, \cGG_m) \cong \ZZ^n$ and $N \da \Hom(\cGG_m, T) \cong \ZZ^n$ where $T\cong (\cGG_m)^n$.
Under the duality pairing, $M \cong N\dual \da \Hom(N, \ZZ)$.
We have $\CC[V] \subseteq \CC[T] = \CC[M] \cong \oplus _{m\in M} \CC z^m$ where $z^{m}da \prod z_i^{m_i}$ when $m = (m_1,\cdots, m_n)$ and $z = (z_1,\cdots, z_n) \in (\cGG_m)^n$.
Similarly for $S \subset M$ we have $\CC[V] = \oplus _{m\in S} \CC z^m$.
For $V$ any affine toric variety $V = \spec \CC[S]$, the associated monoid $S \subseteq M$ has finitely generated.
Recall that if $\sigma \subseteq N_\RR$ is a strongly convex rational polyhedral cone, the dual cone $\sigma\dual = \ts{m\in M_\RR \st \inp m(u) = (m, u) \geq 0 \, \forall u\in \sigma}$.
One can show that \( S_\sigma \da \sigma\dual \intersect M \) is a finitely generated monoid, and defines a variety $V_\sigma \da \spec \CC[S_\sigma]$.
:::

:::{.definition title="Saturated monoids"}
A monoid $S \subseteq M$ is **saturated** iff for all $m\in M$ and $k\in \ZZ_{\geq 1}$, 
\[
km\in S \implies m\in S
.\]
:::

:::{.example title="?"}
Let $V = \spec \CC[x,y]/\gens{x^3-y^2}$, then the associated monoid is $S = \gens{2,3} \subseteq \ZZ$.
For $t\in \cGG_m$, the action is $t.x = t^2x$ and $t.y = t^3 y$.
One can check that $S \cong \NN\smts{1}$, and $S$ is not saturated since $1\in M \cong \ZZ$ and $2\cdot 1 = 2\in S$ but $1\not\in S$.

In contrast, $S \da \NN$ yields a saturated monoid with associated variety $V = \spec \CC[x] = \AA^1\slice \CC$.
:::

:::{.observation}
Monoids of the form $S_\sigma \da \sigma\dual \intersect M$ are always saturated.
Why: such elements $m\in S_\sigma$ are defined by inequalities $\inp{m}{n} \geq 0$, and 
\[
\inp{km}{n}\geq 0 \implies \inp m n \geq 0
.\]
:::

:::{.remark}
Conversely, if $S$ is saturated then $S = S_\sigma$ for some $\sigma$.
Writing $S = \gens{e_1, \cdots, e_k}_{\NN}$, we can construct $\Cone(S) \da \sum_k \RR_{\geq 0} e_k \subseteq M_\RR$ and let $\sigma = \Cone(S)\dual \subseteq N_\RR$.
Note that without saturation, we may not have $S = S_\sigma$, and instead we have 
\[
S_\sigma = \sigma\dual \intersect M = (\Cone(S)\dual)\dual \intersect M = \Cons(S) \intersect M \contains S
,\]
and in fact $S$ is saturated $\iff S = \Cone(S) \intersect M$.
:::

:::{.example title="?"}
Letting $S = \gens{2, 3}$ we have $\Cone(S) = \RR_{\geq 0}$ and thus $1\in \Cone(S) \intersect M$ while $1\not\in S$.
:::

:::{.remark}
Taking the saturation of a cone: given $S$, take $S_\sat \da \Cone(S) \intersect M \contains S$, the smallest saturated monoid containing $S$.
:::

:::{.theorem title="?"}
If $V$ is an affine toric variety associated to $S$, then $V$ is normal $\iff S$ is saturated $\iff S = S_\sigma$ for $\sigma$ an SCRPC in $M$.
:::

:::{.remark}
For any $V$ there is a normalization map $\nu: V^\nu \to V$.
Since $S \subseteq S_\sat$ we have $\CC[S] \injects \CC[S_\sat]$ and thus there is a morphism $\spec \CC[S]\to \spec \CC[S_\sat]$; the claim is that this is the normalization.
:::

:::{.example title="?"}
Consider $V = \spec \CC[x,y]/\gens{x^3-y^2}$ as before with $S = \gens{2,3} \subseteq S_\sat = \NN$, the normalization is 
\[
\nu: \spec \CC[S_\sat] = \spec \CC[\NN] = \AA^1 &\to \spec \CC[S] = \spec \CC[x,y]/\gens{x^3-y^2} \\
t &\mapsto (t^2, t^3)
.\]
:::

:::{.remark}
In dimension 1, the only finitely generated saturated monoids are $\NN, \ZZ$ corresponding to $\AA^1, \GG_m$
:::

:::{.remark}
When is $V$ smooth? It is necessary that $V$ is normal, so $V = \spec \CC[S_\sigma]$ for some $\sigma$, and there exist generators $e_1,\cdots, e_k$ of $\sigma$ which are a subset of a $\ZZ\dash$basis for $N$.
:::

:::{.example title="?"}
In dimension 2, consider first $\sigma = \gens{e_1, e_2}$ compared to $\sigma' = \gens{e_1 + 2e_2, e_1}$. 
The latter does not extend to a $\ZZ\dash$basis since $e_1 + e_2$ is not obtainable as a linear combination of the others.
However, it is normal.
:::