From 7179979353392bce68ac08592531faa1ca383d4b Mon Sep 17 00:00:00 2001
From: "Jiri (George) Lebl" <jiri.lebl@gmail.com>
Date: Fri, 8 Dec 2023 10:37:27 -0600
Subject: [PATCH] Add example and add figure for it, some small fixes

---
 CHANGES                 |   6 +-
 figures/intersect.eepic | 234 ++++++++++++++++++++++++++++++++++++++++
 figures/intersect.xp    |  49 +++++++++
 scv.tex                 |  82 ++++++++------
 4 files changed, 336 insertions(+), 35 deletions(-)
 create mode 100644 figures/intersect.eepic
 create mode 100644 figures/intersect.xp

diff --git a/CHANGES b/CHANGES
index c08ca8f..d4630ae 100644
--- a/CHANGES
+++ b/CHANGES
@@ -174,9 +174,11 @@ and that's more consistent.
 renumbered.
 <li> Simplify the statement in 6.5.4.
 <li> Define the dimension of a subvariety and the concept of pure dimension
-using only regular points, that way we get around the technicality
-that we never prove that the set of regular points is dense in general.
+using only regular points, which seems a little ontologically simpler.
 Also simplifies the last exercise in 6.5.
+<li> Add the intersection of two complex manifolds to example 6.5.6,
+and add the relevant figure.
+Also rename X and Y to U and M so that the naming in the example makes more sense.
 <li> Move the statement of the theorem about \(Z_f\) to section 6.5 from
 6.6 (now Theorem 6.5.9), and add the generalization of this result (that
 regular points exist and are thus dense) to arbitrary varieties,
diff --git a/figures/intersect.eepic b/figures/intersect.eepic
new file mode 100644
index 0000000..9858e52
--- /dev/null
+++ b/figures/intersect.eepic
@@ -0,0 +1,234 @@
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+\draw (0in,0in)--(0in,1.5in);
+\pgftext[at={\pgfpoint{0in}{0in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $\rule{6pt}{0.5pt}$}}}
+\pgftext[at={\pgfpoint{0in}{0.75in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $\rule{6pt}{0.5pt}$}}}
+\pgftext[at={\pgfpoint{0in}{1.5in}}] {\makebox(0,0)[l]{\hbox{\color{rgb_000000}\footnotesize $\rule{6pt}{0.5pt}$}}}
+\pgftext[at={\pgfpoint{-0.055348in}{0in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $-1$}}}
+\pgftext[at={\pgfpoint{-0.055348in}{0.75in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $0$}}}
+\pgftext[at={\pgfpoint{-0.055348in}{1.5in}}] {\makebox(0,0)[r]{\hbox{\color{rgb_000000}\footnotesize $1$}}}
+\pgfsetlinewidth{0.8pt}
+\draw (0in,0in)--(2.25in,1.5in);
+\draw (0in,1.5in)--(2.25in,0in);
+\end{tikzpicture}
diff --git a/figures/intersect.xp b/figures/intersect.xp
new file mode 100644
index 0000000..773cb52
--- /dev/null
+++ b/figures/intersect.xp
@@ -0,0 +1,49 @@
+/* slightly modified sample from epix samples gallery */
+#include "epix.h"
+using namespace ePiX;
+
+int main()
+{
+  double slope;
+  double c;
+  double x;
+
+  picture(P(-1.0,-1.0), P(1.0,1.0), "2.25x1.5in");
+
+  begin();
+
+  //h_axis(5);
+  //v_axis(5);
+
+  //h_axis_labels(3, P(-1, 2), tl); // align top-left
+  //v_axis_labels(3, P(-1, 2), tl);
+
+  pen(Black(0.4));
+  line_style(".");
+  dash_size(1);
+
+  bold();
+  line(P(-1,0),P(1,0));
+  line(P(0,-1),P(0,1));
+
+  plain();
+  line(P(-1,1),P(1,1));
+  line(P(0.5,-1),P(0.5,1));
+  line(P(-0.5,-1),P(-0.5,1));
+  line(P(1,-1),P(1,1));
+
+  solid();
+  pen(Black());
+
+  font_size("footnotesize");
+  bottom_axis(4, P(0,-4)).draw();
+  left_axis(2, P(-4,0)).draw();
+
+  bold();
+  line(P(-1,-1),P(1,1));
+  line(P(-1,1),P(1,-1));
+
+  tikz_format();
+  end();
+}
+
diff --git a/scv.tex b/scv.tex
index 9ce2313..503706e 100644
--- a/scv.tex
+++ b/scv.tex
@@ -15448,13 +15448,13 @@ \section{Varieties} \label{sec:varieties}
 (\exerciseref{exercise:regdimwelldef}).
 
 \begin{example}
-The set $X = \C^n$ is a complex submanifold of dimension $n$
+The set $U = \C^n$ is a complex submanifold of dimension $n$
 (codimension $0$).
-In particular, $X_{\mathit{reg}} = X$ and $X_{\mathit{sing}} = \emptyset$.
+In particular, $U_{\mathit{reg}} = U$ and $U_{\mathit{sing}} = \emptyset$.
 
-The set $Y = \bigl\{ z \in \C^3 : z_3 = z_1^2 - z_2^2 \bigr\}$ is a complex submanifold of
+The set $M = \bigl\{ z \in \C^3 : z_3 = z_1^2 - z_2^2 \bigr\}$ is a complex submanifold of
 dimension $2$ (codimension $1$).  Again,
-$Y_{\mathit{reg}} = Y$ and $Y_{\mathit{sing}} = \emptyset$.
+$M_{\mathit{reg}} = M$ and $M_{\mathit{sing}} = \emptyset$.
 
 On the other hand, the so-called \emph{\myindex{cusp}},
 $C = \bigl\{ z \in \C^2 : z_1^3-z_2^2 = 0 \bigr\}$ is not a complex
@@ -15464,14 +15464,26 @@ \section{Varieties} \label{sec:varieties}
 so $C_{\mathit{reg}} = C \setminus \{0\}$, and so $C_{\mathit{sing}} = \{ 0
 \}$.
 The dimension at every regular point is $1$.
-See \figureref{fig:cusp} for a
+See \figureref{fig:cuspintersect} for a
 plot of $C$ in two real dimensions.
 
+Another type of singularity could be where two complex manifolds
+intersect.  For example, $X = \{ z \in \C^2 : z_1^2-z_2^2 = 0 \}$
+is the union of the two complex manifolds of dimension 1 given by $z_1+z_2=0$ and
+$z_1-z_2=0$.  In this case $X_{\mathit{sing}} = \{ 0 \}$ and
+$X_{\mathit{reg}} = X \setminus \{ 0 \}$.
+See \figureref{fig:cuspintersect} for a
+plot of $X$ in two real dimensions.
+
 \begin{myfig}
 \medskip
 \subimport*{figures/}{cusp.eepic}
+\qquad
+\qquad
+\subimport*{figures/}{intersect.eepic}
 \bigskip
-\caption{The cusp.\label{fig:cusp}}
+\caption{The cusp $C$ (left), and the intersecting manifolds $X$
+(right).\label{fig:cuspintersect}}
 \end{myfig}
 \end{example}
 
@@ -15508,7 +15520,7 @@ \section{Varieties} \label{sec:varieties}
 there is a regular point.
 
 \begin{defn}
-Let $X \subset U \subset \C^n$ be a (complex) subvariety of $U$.  Let $p \in
+Let $X \subset U \subset \C^n$ be a subvariety of $U$.  Let $p \in
 X$ be a point.  We define the (complex)
 \emph{\myindex{dimension}} of $X$ at $p$ to be
 \glsadd{not:dimpX}%
@@ -15542,9 +15554,10 @@ \section{Varieties} \label{sec:varieties}
 of pure dimension $1$.
 \end{example}
 
-Back in \sectionref{sec:riemannextzerosetsinjmaps}, we proved
-following theorem (\thmref{thm:regptsdense}).  We restate it in the
-language of varieties.
+Let us restate
+\thmref{thm:regptsdense} we proved in
+\sectionref{sec:riemannextzerosetsinjmaps}
+in the language of varieties.
 
 \begin{thm} \label{thm:regptsdense2}
 Let $U \subset \C^n$ be a domain and $f \in \sO(U)$.
@@ -15557,7 +15570,7 @@ \section{Varieties} \label{sec:varieties}
 
 Let us improve on this for arbitrary varieties.
 
-\begin{lemma}
+\begin{lemma} \label{lemma:regdense}
 Let $U \subset \C^n$ be open and let $X \subset U$ be a subvariety,
 then $X_{\mathit{reg}}$ is nonempty.  Consequently, $X_{\mathit{reg}}$ is
 open and dense in $X$.
@@ -15573,9 +15586,11 @@ \section{Varieties} \label{sec:varieties}
 $U$, so all points are regular.
 Suppose the lemma is true in dimension $n-1$.
 It is enough to find a regular point in some neighborhood of some point
-$p \in X$, so suppose $p = 0$.  
+$p \in X$.
+Suppose $p = 0$ for simplicity.
 Either $X$ contains a whole neighborhood of $0$, in which case
-$0$ is a regular point, or there is some function $f$ near $0$ that vanishes on $X$.
+$0$ is a regular point, or there is some holomorphic function
+$f$ near $0$ that vanishes on $X$.
 After a small linear change of coordinates, the
 Weierstrass preparation theorem applies and we can assume that $f$ is a
 Weierstrass polynomial.  Write the variables as
@@ -15648,6 +15663,7 @@ \section{Hypervarieties} \label{section:hypervarieties}
 subvariety.
 
 \begin{thm} \label{thm:codim1var}
+\pagebreak[0]
 If $(X,p)$ is a germ of a pure codimension-$1$ subvariety, then
 there is a germ of a holomorphic function $f$ at $p$
 such that $(Z_f,p) = (X,p)$ and $I_p(X)$ is generated by $(f,p)$.
@@ -15682,13 +15698,11 @@ \section{Hypervarieties} \label{section:hypervarieties}
 let $\alpha_1(z'),\ldots,\alpha_k(z')$
 denote the distinct zeros that are in $X'$,
 that is, $\bigl(z',\alpha_\ell(z')\bigr) \in X'$.
-If $\alpha_\ell$ is a holomorphic function in some small neighborhood and
-$\bigl(z',\alpha_\ell(z')\bigr) \in X'$ at one point, then
-$\bigl(z',\alpha_\ell(z')\bigr) \in X'$ for all nearby points too
-as those are clearly in the same component of $X \setminus (E \times D)$.
+Near each point $X'$ is a graph of a holomorphic function over
+$U' \setminus E$, and so we can locally choose 
+$\alpha_1,\ldots,\alpha_k$ to be holomorphic.
 Furthermore, this means that the set $X'$ contains only regular points of
 $X$, which are of dimension $n-1$.
-
 The number of
 such geometrically distinct zeros in $X'$ above each point in
 $U' \setminus E$ is locally constant.
@@ -15726,18 +15740,16 @@ \section{Hypervarieties} \label{section:hypervarieties}
 the fact that $U' \setminus E$ is connected, this means that
 $X \setminus (E \times D)$ has at most finitely many components (at
 most $m$).
-So we can find an $F$ for every topological component of
-$X \setminus ( E \times D )$.  Then we multiply those functions together
-to get $f$.
+We find an $F$ for every topological component of $X \setminus ( E \times D )$
+and we multiply those functions together to get $f$.
 No open piece $M \subset X_{\mathit{reg}}$ can lie completely in $E \times D$,
-as otherwise $M$ would in fact agree with some open piece of $E \times D$,
+as otherwise an open subset of $M$ would also be an open piece of $E \times D$,
 see \exerciseref{exercise:hypersurfaceinhypervariety},
 but we know that $P$ must vanish on $M$, which is impossible as
 it only vanishes at finitely
-many points for each fixed $z'$.  Therefore, as $X_{\mathit{reg}}$
-is dense in $X$, the closure
-of $X \setminus (E \times D)$ contains $X$ and so
-$Z_f = X$.
+many points for each fixed $z'$.
+Therefore, as $X_{\mathit{reg}}$ is dense in $X$ (\lemmaref{lemma:regdense}),
+the closure of $X \setminus (E \times D)$ contains $X$ and so $Z_f = X$.
 
 The fact that this $f$ generates $I_p(X)$ is left as
 \exerciseref{exercise:singlegenerator}.
@@ -15748,8 +15760,8 @@ \section{Hypervarieties} \label{section:hypervarieties}
 
 \begin{example}
 It is not true that
-if a dimension of a subvariety in $\C^n$ is $n-k$ (codimension $k$),
-there are $k$
+a subvariety in $\C^n$ of dimension $n-k$ (codimension $k$)
+has $k$
 holomorphic functions that ``cut it out.''  That only works for $k=1$.
 The set defined by
 \begin{equation*}
@@ -15824,12 +15836,16 @@ \section{Hypervarieties} \label{section:hypervarieties}
 \end{exercise}
 
 \begin{exercise} \label{exercise:hypersurfaceinhypervariety}
-Suppose that $U \subset \C^n$ is a domain and $X \subset U$
-is a subvariety of dimension $n-1$.  Suppose that $M$
+Suppose $U \subset \C^n$ is open and $X \subset U$
+is a subvariety of dimension $n-1$.  Suppose $M$
 is a small piece of a complex submanifold of dimension $n-1$ such that
-$M \subset X$.  Prove that the set $M \cap X_{\textit{sing}}$ is
-nowhere dense in $M$.  Hint: Locally near some point of $M$, make $M$ into
-$\{z_n = 0 \}$ and apply \thmref{thm:discrthm} to $X$ there.
+$M \subset X$.  Prove that $M$ agrees with $X_{\textit{reg}}$ on a dense
+open set, that is,
+for each $p$ a dense open subset of $M$,
+there is a neighborhood $W$ of $p$ such that
+$M \cap W = X_{\textit{reg}} \cap W$.
+Hint: Consider coordinates where
+$M$ is a graph and \thmref{thm:discrthm} applies to $X$.
 \end{exercise}
 
 \begin{exercise}