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\title{
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    Preview
  }
    \\ {\normalsize University of Georgia} \\
  }







\begin{document}

\date{}
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\begin{flushleft}
\textit{D. Zack Garza} \\
\textit{University of Georgia} \\
  \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\
{\tiny \textit{Last updated:} 2023-12-17 }
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\hypertarget{wednesday-august-16}{%
\section{Wednesday, August 16}\label{wednesday-august-16}}

\begin{remark}

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

\end{remark}

\begin{remark}

Define \(S \coloneqq{\mathbf{C}}[x_1, \cdots, x_n]\), and an ``affine
algebraic variety'' to be the common vanishing locus in
\({\mathbf{C}}^n\) of a system of equations. Say two such varieties
\(C_1, C_2\) are isomorphic iff
\({\mathbf{C}}[C_1] \cong {\mathbf{C}}[C_2]\) as
\({\mathbf{C}}{\hbox{-}}\)algebras, where
\({\mathbf{C}}[C_i] \coloneqq S / I(C_i)\) and
\(I(C_i) \coloneqq\left\{{f\in S {~\mathrel{\Big\vert}~}f(p) = 0 \,\, \forall p\in C_i}\right\}\).

\end{remark}

\begin{exercise}[?]

Let \(V\) be an affine variety and show that
\(I(V) \coloneqq\left\{{ f\in S {~\mathrel{\Big\vert}~}f(p) = 0 \,\,\forall p\in V}\right\}\)
is in fact an ideal.

\end{exercise}

\begin{exercise}[?]

Show that
\(\left\{{\text{points of }V}\right\}\rightleftharpoons\operatorname{mSpec}{\mathbf{C}}[V]\)
where \(p\rightleftharpoons{\mathfrak{m}}_p\) is a bijection.

\end{exercise}

\begin{fact}

\(V\) is irreducible iff \({\mathbf{C}}[V]\) is an integral domain iff
\(I(V)\) is a prime ideal.

\end{fact}

\begin{remark}

For dimension 1, normal is equivalent to smooth, but in higher
dimensions.

\end{remark}



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