# Wednesday, August 16

:::{.remark}
See the [syllabus](https://drive.google.com/file/d/1uWvSFqhMZU8lHDpNxRY7sKW188xzu4tH/view?pli=1) and [calendar](https://drive.google.com/file/d/19hh7iTG8ZFx7uYHCmWU-Db2cbVFZNlud/view).
Textbooks: Fulton's *Toric Varieties* and Cox-Little-Schenck's book of the same title. 
We'll also consider Gross-Hacking-Keel-Kontsevich's paper [Canonical bases for cluster algebras](https://arxiv.org/abs/1411.1394).

:::

:::{.remark}
Define $S \da \CC[x_1, \cdots, x_n]$, and an "affine algebraic variety" to be the common vanishing locus in $\CC^n$ of a system of equations.
Say two such varieties $C_1, C_2$ are isomorphic iff $\CC[C_1] \cong \CC[C_2]$ as $\CC\dash$algebras, where $\CC[C_i] \da S / I(C_i)$ and $I(C_i) \da \ts{f\in S \st f(p) = 0 \,\, \forall p\in C_i}$.
:::

:::{.exercise title="?"}
Let $V$ be an affine variety and show that $I(V) \da \ts{ f\in S \st f(p) = 0 \,\,\forall p\in V}$ is in fact an ideal.
:::

:::{.exercise title="?"}
Show that $\ts{\text{points of }V}\mapstofrom \mspec \CC[V]$ where $p\mapstofrom \mfm_p$ is a bijection.
:::

:::{.fact}
$V$ is irreducible iff $\CC[V]$ is an integral domain iff $I(V)$ is a prime ideal.
:::

:::{.remark}
For dimension 1, normal is equivalent to smooth, but in higher dimensions.
:::