# Monday, August 21

## 1.1 (continued)

> Todo: missed 15-20m!

:::{.remark}
Consider $S \da \gens{2, 3}_{\ZZ_{\geq 0}} = \ts{2a + 3b \st a,b,\in \ZZ_{\geq 0}}$.
Any $S \embeds M$ yields a variety $X_S \da \mspec \CC[S]$.
In this case, $X_S = \mspec \CC[x,y]/\gens{x^3-y^2}$.
Any $m\in M$ defines a function $\chi^m: T_t \to \cstar$ (i.e. a cocharacter) where $T_t\cong \cstar$, where e.g. $\chi^m(t) = t^m$ or 
\[
\chi^{(3, 5)}: (\cstar)^2 &\to \cstar \\
(s, t) &\mapsto s^3 t^5
.\]

Here $\CC[S] = \ts{ \chi^0, \chi^2, \chi^3, \chi^{4}, \chi^5, \cdots }$, which is the ring of regular functions on $X_S$.
Moreover $\mspec \CC[S] = \mspec \CC[x,y]/\gens{x^3-y^2}$.
Note that there is a morphism
\[
\phi: \NN^2 &\to S \\
(1, 0) &\mapsto 2 \\
(0, 1) &\mapsto 3
,\]
which induces 
\[
\phi^*: \CC[\NN^2] &\to \CC[S] \\
x = z^{(1, 0)} &\mapsto t^2 \\
y = z^{(0, 1)} &\mapsto t^3
.\]
One can check $\ker \phi^* = \gens{x^3-y^2}$ and thus $\CC[\NN^2]/\gens{x^3-y^2}$.
:::

## 1.2

:::{.remark}
Goal: construct affine toric varieties from cones in $N_\RR$ (the cocharacter lattice).
A rational polyhedral cone in $N_\RR$: given finitely many elements in $N$, take the collection of non-negative real linear combinations.
E.g. $\gens{e_0, e_1}_{\RR_{\geq 0}}$ yields a cone $\Cone(e_1, e_2) = \ts{x\geq 0, y\geq 0}$, i.e. the first quadrant.
Say a cone $\Cone(v_1,\cdots, v_n)$ is rational if $v_i\in N$ for all $i$, and polyhedral if there are only finitely many such vectors.

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:::

:::{.remark}
To any $\sigma =\Cone(v_1,\cdots, c_n) \subseteq N_\RR$, associate 
\[
\sigma\dual \da \ts{m \in M \mid \inp m {v_i} \geq 0 \,\,\forall v_i }
.\]
See $\S 1.2$ for how to find generators of $\sigma\dual$ given generators of $\sigma$.

For $\ts{e_1, e_2}$ spanning $N$, there is a dual basis $\ts{e_1\dual, e_2\dual}$ such that $\inp {e_i}{e_j\dual} = \delta_{ij}$.
Here $\sigma\dual = \gens{e_1\dual, e_2\dual}_{\RR_{\geq 0}}$, and we get two conditions: 
\[
\inp{Ae_1\dual + Be_2\dual}{e_1} &\geq 0 \implies A\geq 0 \\
\inp{Ae_1\dual + Be_2\dual}{e_2} &\geq 0 \implies B\geq 0
.\]
In general take $S_\sigma = \sigma\dual \intersect M$ the monoid of integral points in the dual cone and $U_\sigma = \spec \CC[S_\sigma]$.
By definition $S_\sigma$ is saturated and thus $U_\sigma$ is always normal, and in fact all normal toric varieties are obtainable by this procedure.
:::