# Friday, August 18

:::{.exercise title="?"}
Let $V \da \spec \CC[x_1, x_2]/\gens{x_2-x_1^2}$ and show $V\cong \AA^1\slice \CC$. 
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:::{.exercise title="?"}
Show that there are varieties $V_1, V_2$ with coordinate rings that are isomorphic as rings but not as $\CC\dash$algebras, and thus $V_1\not\cong V_2$.

> Hint: see Serre, *Exemples de variétés projectives conjuguées non homéomorphes*.

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:::{.example title="?"}
Check that $(\cstar)^n \injects \CC$ is a variety via the presentation $V(\prod_{i=1}^{n+1} - 1)$, defining it as a variety.
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:::{.remark}
A character of a torus $T \da (\cstar)^n$ is a morphism $\chi: T\to \cstar$.
For any $m = (a_1,\cdots, a_n) \in \ZZ^n$, there is a character $\chi^m$ given by $(t_1,\cdots, t_n) \mapsto \prod t_i^{a_i}$.
Define $M$ to be the set of all characters of $T$.
:::

:::{.remark}
If $T$ acts linearly on a vector space $W$, there is an eigenspace decomposition $W = \bigoplus_{m\in M} W_m$ where $W_m \da \ts{w\in W \st t.w = \chi^m(t) w\, \forall t\in T}$ is the weight space of weight $m$.
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:::{.remark}
A cocharacter is a group morphism $\lambda: \CC^* \to T$ and any $u=(b_1,\cdots, b_n) \in \ZZ^n$ where \( \lambda^u(t) \da (t^{b_1}, \cdots, t^{b_n}) \).
All cocharacters are of this form, and we define $N$ to be the set of all cocharacters of $T$.
Note that $M$ and $N$ are dual lattices under the pairing
\[
\inp{\wait}{\wait}: M\times N &\to \ZZ \\
( \chi, \lambda) &\mapsto \ell
\]
where $\ell$ is a unique integer determined in the following way: note that \( \chi \circ \lambda \) is a map $\cstar\to\cstar$ and thus of the form $z\mapsto z^\ell$.
:::

:::{.definition title="Affine toric varieties"}
An **affine toric variety** is an irreducible variety[^normality] $V$ containing an open dense torus $T\da (\cstar)^n$ such that the action of $T$ on itself extends to an action on $V$.

[^normality]: 
Note that Fulton imposes the condition of normality, which we'll not impose here yet.

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:::{.exercise title="?"}
Show that $(\cstar)^n$ and $\CC^n$ are affine toric varieties.
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:::{.exercise title="?"}
Show that $V(x^3-y^2) \subseteq \CC^2$ is an affine toric variety with an embedding 
\[
\cstar &\embeds \CC^2 \\
t &\mapsto (t^2, t^3)
.\]

Show that the action $\cstar \selfmap$ extends to an action on $V$.

> Hint: look at the following map,
\[
\cstar \times V &\to V \\
t.x &\da t^2 x \\
t.y &\da t^3 y
.\]

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:::{.remark}
Note that since $T \subseteq V$ for a toric variety $V$, we have $\CC[V] \subseteq \CC[T] \cong \CC[x_1,\cdots, x_{n+1}]/\gens{\prod x_i - 1} \cong \CC[x_1^{\pm 1}, \cdots, x_n^{\pm 1}]$ which admits a decomposition $\CC[T] \cong \oplus _{m\in M} \CC \chi^m$. I.e., any Laurent polynomial can be written as a sum of monomials of the form $\prod x_i^{a_i}$ for $m = (a_1,\cdots, a_n)$. To any $V$ there is an associated monoid $S \subseteq M$ such that $\CC[V] \cong \oplus _{m\in S} \CC \chi^m$.
Conversely, for each finitely generated monoid $S \embeds M$, there is an associated toric variety $M$.
One can write a monoid algebra 
\[
\CC[S] \da \ts{ \sum _{m\in S} c_m \chi^m \st c_m \in \CC, c_m =0 \text{ for all but finitely many }m}
.\]
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:::{.exercise title="?"}
Set $S = M = \ZZ^n$, then $\spec \CC[M] \cong (\cstar)^n$.
Similarly, if $S = \NN^n \subseteq M$ then $\spec \CC[S] \cong \CC^n$.
Check that if $S = \gens{2e_1, 3e_2}$ then $\spec \CC[S] \cong V(x^3-y^2)$.
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