Some improvements / fixes

scv.tex

@@ -15579,7 +15579,7 @@ \section{Varieties} \label{sec:varieties}
 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
-$(z_1,\ldots,z_{n-1},z_n) = (z',z_n)$ as before.
+$(z_1,\ldots,z_{n-1},z_n) = (z',z_n)$.
 In some neighborhood $V$ of the origin (where $V \subset U$),
 \thmref{thm:discrthm} applies to $f$, so
 let $\Delta(z')$ denote the discriminant function.
@@ -15597,12 +15597,14 @@ \section{Varieties} \label{sec:varieties}
 After a linear change of variables in $z'$,
 we apply the preparation theorem in $z'$ with respect to $z_{n-1}$
 and get $\Delta = u P$ in some neighborhood of $0$,
-where $u$ is a unit $P$ is a Weierstrass polynomial
-in $z_{n-1}$ with coefficients that only depend on $z_1,\ldots,z_{n-2}$.
+where $u$ is a unit and $P$ is a Weierstrass polynomial
+in $z_{n-1}$ with coefficients that only depend on
+$z_1,\ldots,z_{n-2}$.
 Let $\Delta'$ be the discriminant for $P$ and repeat the procedure
-above.  Either $\Delta'$ does not vanish some point of $X$, in which case
-we apply the induction hypothesis as above, or near the
-origin $X$ is contained in the zero set of $\Delta'$.  If $X$ is contained
+above.  Either $\Delta'$ is nonzero at some point of $X$,
+in which case we apply the induction hypothesis as above,
+or near the
+origin, $X$ is contained in the zero set of $\Delta'$.  If $X$ is contained
 in the zero set of $\Delta'$, we again apply the preparation theorem
 and get a polynomial in $z_{n-2}$ with coefficients depending only on
 $z_1,\ldots,z_{n-3}$.  Rinse and repeat.  Either at some point we could
@@ -15620,7 +15622,7 @@ \section{Varieties} \label{sec:varieties}
 \begin{thm} \label{thm:varietysingularity}
 Let $U \subset \C^n$ be open and let $X \subset U$
 be a subvariety, then
-$X_{\mathit{sing}} \subset X$ is a subvariety
+$X_{\mathit{sing}} \subset X$ is a subvariety,
 which is nowhere dense in $X$
 and $\dim X_{\mathit{sing}} < \dim X$.
 \end{thm}
@@ -15652,29 +15654,33 @@ \section{Hypervarieties} \label{section:hypervarieties}
 \end{thm}
 
 \begin{proof}
-We need to find a function that vanishes on $(X,p)$ and divides every other
-function that vanishes there.
-There has to exist at least one germ of a function that vanishes on $(X,p)$
+We need to find a function that vanishes on $(X,p)$
+and divides every other function that vanishes there.
+There must exist at least one germ of a function that vanishes
+on $X$ near $p$
 (although it could vanish on a larger set).
-Assume $p=0$, and after a linear change
-of coordinates assume we can apply the Weierstrass preparation theorem
-to the function.  Taking representatives of the germs, we assume
-$X$ is a pure codimension-$1$ subvariety of a small enough
+Without loss of generality, assume $p=0$
+and after a linear change
+of coordinates the Weierstrass preparation theorem applies.
+More precisely,
+suppose $X$ is a pure codimension-$1$ subvariety
+of a small enough
 neighborhood
 $U' \times D \subset \C^{n-1} \times \C$ of the origin, where $D$ is a disc,
 and the function that vanishes on $X$ is a
 Weierstrass polynomial $P(z',z_n)$ defined for $z' \in U'$, and
 all zeros of $z_n \mapsto P(z',z_n)$ are in $D$ for $z' \in U$.
-
 \thmref{thm:discrthm} applies. Let
 $E \subset U'$ be the discriminant set, a zero set of a
 holomorphic function.
 On $U' \setminus E$, there are a certain number of geometrically
 distinct zeros of $z_n \mapsto P(z',z_n)$.
 
-Let $X'$ be a topological component of $X \setminus ( E \times D )$.
-Above each point $z' \in U' \setminus E$, let
-$\alpha_1(z'),\ldots,\alpha_k(z')$ denote the distinct zeros that are in $X'$,
+Let $X'$ be a topological component of
+$X \setminus ( E \times D )$.
+Above each $z' \in U' \setminus E$,
+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
@@ -15687,7 +15693,7 @@ \section{Hypervarieties} \label{section:hypervarieties}
 such geometrically distinct zeros in $X'$ above each point in
 $U' \setminus E$ is locally constant.
 As $U' \setminus E$ is connected
-(\exerciseref{exercise:connectedcomplement})
+(\exerciseref{exercise:connectedcomplement}),
 there exists a unique $k$.  Take
 \begin{equation*}
 F(z',z_n) = \prod_{\ell=1}^k \bigl( z_n-\alpha_\ell(z')\bigr)
@@ -15697,18 +15703,21 @@ \section{Hypervarieties} \label{section:hypervarieties}
 The coefficients $g_\ell$ are well-defined for $z \in U' \setminus E$
 as they are independent of how $\alpha_1,\ldots,\alpha_k$ are ordered.
 The $g_\ell$ are holomorphic for $z \in U' \setminus E$
-as locally we can choose the order so that each $\alpha_\ell$ is
-holomorphic.
+as locally we can ensure that each $\alpha_\ell$ is holomorphic.
 The coefficients $g_\ell$ are bounded
-on $U'$ and therefore extend to holomorphic functions of $U'$.
-Hence, the polynomial $F$ is a polynomial
+on $U'$ and so extend to holomorphic functions of $U'$
+via the Riemann extension theorem.
+Hence, $F$ is a polynomial
 in $\sO(U')[z_n]$.
 The zeros of $F$
 above $z' \in U' \setminus E$
 are simple and give precisely $X'$.
+The zeros of $F$ above $z' \in E$, must be
+limits zeros above points of $U' \setminus E$
+by the argument principle.
 Consequently,
-the zero set of $F$ is the closure of $X'$ in $U' \times D$ by continuity.
-It is left to the reader to check that %(using the argument principle)
+the zero set of $F$ is the closure of $X'$ in $U' \times D$.
+It is left to the reader to check that
 all the functions $g_\ell$ vanish at the origin and $F$ is a Weierstrass
 polynomial, a fact that will be useful in the exercises below.
 
@@ -15887,26 +15896,29 @@ \section{Hypervarieties} \label{section:hypervarieties}
 $(X,p)$ meaning that its zero set is equal to $X$ near $p$.
 Without loss of generality, assume that $p=0$,
 and after a linear change of coordinates, assume that we can apply
-the preparation theorem and \thmref{thm:discrthm} near the origin with respect to each variable
-(e.g.\ if we can apply the preparation in each variable).
+the preparation theorem and \thmref{thm:discrthm}
+near the origin with
+respect to each variable.
 Then we can assume that $f$ is holomorphic in some neighborhood $W$,
 $Z_f = X \cap W$, and
 there exists an open neighborhood $V$ of the origin so that for every
 variable $z_k$, $k=1,\ldots,n$, there is a polydisc $D=D_1 \times \cdots
 \times D_n$ centered at the origin with
 $\widebar{V} \subset D$ and $\widebar{D} \subset W$, where
 \thmref{thm:discrthm} applies with respect to $z_k$, that is,
-$Z_f \cap D_1 \times \cdots \times \partial D_k \times \cdots \times D_n =
-\emptyset$.
+$Z_f \cap
+(D_1 \times \cdots \times \partial D_k \times \cdots \times D_n)
+= \emptyset$.
 
 Consider a $q \in X_{\textit{reg}} \cap V$.  By definition, $X$ is a graph
-near $q$, so we can, after reordering variables, assume it is a graph of $z_n$
+near $q$, so after reordering variables,
+we assume it is a graph of $z_n$
 over $z'=(z_1,\ldots,z_{n-1})$.
 Let $D$ be the corresponding polydisc and
 write $D = D' \times D_n \subset \C^{n-1} \times \C$.  Let $E \subset D'$
 be the discriminant set given by the function $\Delta \in \sO(D')$.
-We can think of $\Delta$ as a function in $\sO(D)$.
-If $q \not\in E \times D_n$, then we have $\Delta(q) \not= 0$, so we have
+We think of $\Delta$ as a function in $\sO(D)$.
+If $q \not\in E \times D_n$, then $\Delta(q) \not= 0$, so we have
 found a function holomorphic in $V$ that is nonzero at $q$.
 Let us start a collection $\sF$ of holomorphic functions on $V$,
 one for each $q \in X_{\mathit{reg}} \cap V$,
@@ -15948,20 +15960,23 @@ \section{Hypervarieties} \label{section:hypervarieties}
 $\epsilon'$ will give us $k$ distinct intersections nearby.
 See \exerciseref{exercise:nonverticalintersections}.
 
-We take a slightly smaller polydisc
-$\widetilde{D} \subset \widetilde{D}' \times D_n \subset D$
+Take a slightly smaller polydisc
+$\widetilde{D} = \widetilde{D}' \times D_n \subset D$
 in the $\tilde{z}$ variables
 such that
 still $V \subset \widetilde{D}$ (we may need to pick $\epsilon'$ small
-enough to arrange this), we can apply \thmref{thm:discrthm}.  As the number
+enough to arrange this)
+and \thmref{thm:discrthm} applies in $\widetilde{D}$.
+As the number
 of distinct zeros of $f\bigl(\tilde{z}' + (q_n-z_n) \epsilon', z_n\bigr)$ is $m$
 for all $\tilde{z}'$ near $q'$ including $\tilde{z}'=q'$,
-we find that the discriminant
+the discriminant
 $\widetilde{\Delta}$ in these variables does not vanish at $q'$.
 We againt consider $\widetilde{\Delta}$ as a function
 on $V$ and as it does not vanish at $q$, we add $\widetilde{\Delta}$ to $\sF$.
-See \figureref{fig:Xsingvariety} for the setup.  We have finished defining
-the $\sF$.
+See \figureref{fig:Xsingvariety} for the setup.
+We define $\sF$ by repeating for each
+$q \in X_{\mathit{reg}} \cap V$.
 
 \begin{myfig}
 \medskip
@@ -15979,7 +15994,8 @@ \section{Hypervarieties} \label{section:hypervarieties}
 Thus the common zero set of all the functions in $\sF$ intersected with
 $X \cap V$ gives us precisely $X_{\textit{sing}} \cap V$, so
 $X_{\textit{sing}}$ is a subvariety.  It cannot be of dimension $n-1$ as
-if it were it would be a complex submanifold of dimension $n-1$ near some
+if it were,
+it would be a complex submanifold of dimension $n-1$ near some
 point and then not all of those points would be singular for $X$,
 see \exerciseref{exercise:hypersurfaceinhypervariety}.
 \end{proof}
@@ -15993,7 +16009,8 @@ \section{Hypervarieties} \label{section:hypervarieties}
 $n$ discriminant functions, one for each variable.
 More than $n$ functions
 may be necessary as situations like the one depicted in
-\figureref{fig:Xsingvariety} may occur for some $q$ no matter how we change
+\figureref{fig:Xsingvariety} may occur for some $q$,
+no matter how we change
 variables to start with.
 
 \begin{exbox}
@@ -16547,7 +16564,7 @@ \section{Segre varieties and CR geometry} \label{sec:crgeomcr}
 
 \begin{proof}
 Let $U$ be a neighborhood of $p$ where a representative
-of $\Sigma_p$ is defined, that is, we assume that $\Sigma_p$ is
+of $\Sigma_p$ is defined, that is, assume that $\Sigma_p$ is
 a closed subset of $U$, and suppose $r(z,\bar{z})$ is the corresponding
 defining function.
 Let $\varphi \colon \D \to \C^n$ be the parametrization of $\Delta$
@@ -16556,7 +16573,7 @@ \section{Segre varieties and CR geometry} \label{sec:crgeomcr}
 is sufficient (if there are multiple points of $\D$
 that go to $p$, we repeat the argument for each one).
 So let us assume without loss of generality that $\varphi(\D) = \Delta \subset U$.
-Since $\Delta \subset M$ we have
+Since $\Delta \subset M$, we have
 \begin{equation*}
 r\bigl(\varphi(\xi),\overline{\varphi(\xi)}\bigr) =
 r\bigl(\varphi(\xi),\bar{\varphi}(\bar{\xi})\bigr) = 0 .