Add lecture 4

lecture-notes/lecture-notes.tex

@@ -1,5 +1,6 @@
 \documentclass[a4paper,12pt]{amsart}
 
+\newcommand{\ZZ}{\mathbb{Z}}
 \newcommand{\RR}{\mathbb{R}}
 \newcommand{\CC}{\mathbb{C}}
 \newcommand{\PP}{\mathbb{P}}
@@ -17,6 +18,7 @@
 		-I
 \end{bmatrix}}
 \DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Spec}{Spec}
 
 \usepackage[top=1in, bottom=1in, left=1in, right=1in]{geometry}
 \usepackage{amsmath, amssymb}
@@ -619,4 +621,92 @@ \section{Lecture 03}
 
 The set $H$ from the above proposition is the Hilbert basis of $S_\sigma$.
 
+%---- Lecture 04 on Feb 5 ---------
+\newpage
+\section{Lecture 04}
+
+Let $\sigma$ be a cone in $N$ (i.e., $\sigma$ is a rational, polyhedral, convex cone in $N_\mathbb{R}$).
+Often $\sigma$ is pointed.
+Recall that $S_\sigma = \sigma^\vee \cap M \subseteq M$ ($= \mathbb{Z}^n$) is a finitely generated semigroup by its Hilbert basis.
+
+\begin{Def}
+Let $A_\sigma = \CC[S_\sigma] = \CC[\sigma^\vee \cap M]$.
+This is the $\CC$-algebra such that
+\begin{enumerate}
+\item $\{t^m : m \in S_\sigma\}$ is a $\CC$-basis for $A_\sigma$
+\item $t^{m_1} \cdot t^{m_2} = t^{m_1 + m_2}$
+\end{enumerate}
+\end{Def}
+
+\begin{Def}
+$X_\sigma = \Spec A_\sigma$ is the \emph{affine toric variety} associated to $\sigma$.
+\end{Def}
+
+We want to understand:
+\begin{itemize}
+\item examples
+\item ideal
+\item points
+\item tori and torus action
+\item tangent space and singularities
+\item toric morphisms.
+\end{itemize}
+
+Note that if $\sigma^\vee \cap M = \langle m_1, \dots, m_r \rangle$ then $A_\sigma$ is generated by $\{t^{m_1}, \dots, t^{m_r}\}$.
+In particular, $A_\sigma$ is Noetherian.
+We have an exact sequence
+\[
+0 \to \ker \phi \to \CC[x_1, \dots, x_r] \overset{\phi}{\to} A_\sigma \to 0
+\]
+where $\phi: x_i \mapsto t^{m_i}$.
+Then defining $I_\sigma = \ker \phi$ gives that $A_\sigma = \CC[x_1, \dots, x_r]/I_\sigma$.
+Thus $X_\sigma = V(I_\sigma) \subseteq \CC^r$ is an affine variety.
+
+\begin{Eg}
+Let $\sigma = \mathrm{vcone}\left(\begin{pmatrix}-1\\0\end{pmatrix}, \begin{pmatrix}0\\-1\end{pmatrix}\right)$.
+
+\includegraphics[width=0.3\textwidth]{pic/lec04-pic1}
+\includegraphics[width=0.3\textwidth]{pic/lec04-pic2}
+
+Then $A_\sigma = \CC[s, t]$ and $X_\sigma = \CC^2$.
+\end{Eg}
+
+\begin{Eg}
+If $\sigma = \mathrm{vcone}(-I_n) \subseteq \RR^n$ then $X_\sigma = \CC^n$.
+If $\sigma = \mathrm{vcone}(I_n)$ then $X_\sigma = \CC^n$ but with different coordinates.
+\end{Eg}
+
+\begin{Eg}
+Let $\sigma = \{0\}$ in $N = \ZZ^2$.
+Then $\sigma^\vee = M_\RR$, $S_\sigma$ is all lattice points, and $A_\sigma = \CC[s,t,s^{-1},t^{-1}] = \CC[s, t]_{st}$.
+This gives $X_\sigma = (\CC^\star)^2$, which is the algebraic torus $T$.
+\end{Eg}
+
+\begin{Eg}
+Let $\sigma = \{0\}$ in $N = \ZZ^n$.
+Then $\sigma^\vee = M_\RR$, $S_\sigma$ is all lattice points, and $A_\sigma = \CC[t_1^{\pm 1}, \dots, t_n^{\pm 1}]$.
+This gives $X_\sigma = (\CC^\star)^n$, which is often denoted $T_n$.
+\end{Eg}
+
+\begin{Eg}
+Let $\sigma = \mathrm{vcone}\left(\begin{pmatrix}-1\\2\end{pmatrix}, \begin{pmatrix}0\\-1\end{pmatrix}\right)$.
+Then $\sigma^\vee = \mathrm{vcone}\left(\begin{pmatrix}2\\1\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix}\right)$.
+
+\includegraphics[width=0.3\textwidth]{pic/lec04-pic3}
+
+We have $A_\sigma = \CC[s, s^2t] = \CC[u, v]$ and $X_\sigma = \CC^2$.
+
+Now let $\tau = \mathrm{vcone}\left(\begin{pmatrix}-1\\2\end{pmatrix}\right)$ so $\tau^\vee = \mathrm{vcone}\left(\begin{pmatrix}2\\1\end{pmatrix}, \begin{pmatrix}-2\\1\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix}\right)$.
+Then $A_\tau = \CC[s^2t, (s^2t)^{-1}, s] = (A_\sigma)_{s^2t}$ and $X_\tau = \CC \times \CC^\star \hookrightarrow X_\sigma$ as an open set.
+\end{Eg}
+
+Exercise: If $\sigma$ is not pointed, what is $A_\sigma$ and $X_\sigma$?
+
+As before, let $\sigma^\vee \cap M = \langle m_1, \dots, m_r \rangle$ and consider the exact sequence
+\[
+0 \to I_\sigma \to \CC[x_1, \dots, x_r] \overset{\phi}{\to} A_\sigma \to 0.
+\]
+Note that $A_\sigma = \CC[\sigma^\vee \cap M] \subseteq \CC[M] = \CC[t_1^{\pm 1}, \dots, t_n^{\pm 1}]$ and $\CC[t_1^{\pm 1}, \dots, t_n^{\pm 1}]$ is a domain.
+Thus $A_\sigma$ is a domain, $I_\sigma$ is prime, and $X_\sigma$ is irreducible.
+
 \end{document}