fix: proofread Algebraic Geometry I part

tex/alg-geom/affine-var.tex

@@ -413,7 +413,7 @@ \section{Pictures in $\Aff^2$ and $\Aff^3$}
 	The vanishing set of this is $\VV(x-a, y-b) = \{ (a,b) \} \in \CC^2$,
 	so these ideals correspond to a single point.
 	\ii Let $f(x,y)$ be an irreducible polynomial, like $y-x^2$.
-	Then $(f)$ is a prime ideal! Here $\VV(I)$ is a ``degree one curve''.
+	Then $(f)$ is a prime ideal! Here $\VV(I)$ is a ``codimension one curve''.
 \end{itemize}
 
 By using some polynomial algebra
@@ -442,7 +442,7 @@ \section{Pictures in $\Aff^2$ and $\Aff^3$}
 
 The picture in $\Aff^3$ is harder to describe.
 Again, you have points $\VV(x-a, y-b, z-c)$ corresponding to
-be zero-dimensional points $(a,b,c)$, and two-dimensional surfaces
+zero-dimensional points $(a,b,c)$, and two-dimensional surfaces
 $\VV(f)$ for each irreducible polynomial $f$ (for example, $x+y+z=0$ is a plane).
 But there are more prime ideals, like $\VV(x,y)$, which corresponds to the
 intersection of the planes $x=0$ and $y=0$: this is the one-dimensional $z$-axis.
@@ -549,7 +549,7 @@ \section{Motivating schemes with non-radical ideals}
 
 In order to encapsulate this, we need a more refined object than
 a variety, which (at the end of the day) is just a set of points;
-it's not possible using topology along to encode more information
+it's not possible using topology alone to encode more information
 (there is only one topology on a single point!).
 This refined object is the \emph{scheme}.
 

tex/alg-geom/bezout.tex

@@ -226,7 +226,7 @@ \section{Hilbert functions of finitely many points}
 	To add in the $m$th point we note that
 	\[
 		h_{I_1 \cap \dots \cap I_m}(d)
-		= h_{I_1 \cap \dots I_{m-1}}(d) + h_{I_m}(d)
+		= h_{I_1 \cap \dots \cap I_{m-1}}(d) + h_{I_m}(d)
 		- h_J(d)
 	\]
 	where $J = (I_1 \cap \dots \cap I_{m-1}) + I_m$.
@@ -273,7 +273,7 @@ \section{Hilbert polynomials}
 		h_{(zy-x^2)}(1) &= 3 & \text{Basis $x,y,z$} \\
 		h_{(zy-x^2)}(2) &= 5 & \text{Basis $xy$, $xz$, $y^2$, $yz$, $z^2$}.
 	\end{align*}
-	We thus in fact see that $h_{(zy-x^2)}(d) = 2d-1$.
+	We thus in fact see that $h_{(zy-x^2)}(d) = 2d+1$.
 \end{example}
 
 In fact, this behavior of ``eventually polynomial'' always works.
@@ -404,7 +404,7 @@ \section{B\'ezout's theorem}
 \end{theorem}
 Geometrically,
 \begin{moral}
-	If $V$ is any projective variety, $\VV(f)$ is a hyperplane of degree $k$, then their
+	If $V$ is any projective variety, $\VV(f)$ is a hypersurface of degree $k$, then their
 	intersection $V \cap \VV(f)$ has degree $k \deg V$ --- unless
 	some irreducible component of $V$ is contained inside $\VV(f)$.
 \end{moral}
@@ -516,7 +516,7 @@ \section{Applications}
 	so they must lie on this line.
 \end{proof}
 
-We'd like to remark that the Pascal's theorem is just a special case of the Cayley-Bacharach theorem,
+We'd like to remark that Pascal's theorem is just a special case of the Cayley-Bacharach theorem,
 which can be used to prove that the addition operation on an elliptic curve is associative.
 Interested readers may want to try proving the Cayley-Bacharach theorem using the same technique.
 
@@ -531,7 +531,7 @@ \section{Applications}
 		\[ \chi_{I+(f)}(d) = \chi_I(d) - \chi_I(d-k). \]
 		Let $m = \dim \Vp(I) \ge 1$.
 		Now $\dim \Vp(I+(f)) = m-1$, so
-		and $c_{\text{new}} = \deg I+(f)$ then we have
+		we have
 		\[
 			\frac{\deg (I+(f)) d^{m-1} + \dots}{(m-1)!}
 			=

tex/alg-geom/proj-var.tex

@@ -138,7 +138,7 @@ \section{The ambient space}
 	= \lambda^d f(x_0, \dots, x_n).
 \]
 In particular, the relation ``$f(x_0, \dots, x_n) = 0$'' is
-well-defined if $F$ is homogeneous. Thus, we can say:
+well-defined if $f$ is homogeneous. Thus, we can say:
 \begin{definition}
 	If $f$ is homogeneous, we can then define its \vocab{vanishing locus} as
 	\[
@@ -232,7 +232,7 @@ \section{Homogeneous ideals}
 	be projective varieties in $\CP^n$.
 	We can consider them as \emph{affine varieties} in $\Aff^{n+1}$
 	by using the interpretation of $\CP^n$
-	as lines through the origin in $\CC^n$.
+	as lines through the origin in $\CC^{n+1}$.
 
 	Algebraically, this is done by taking the homogeneous ideals
 	$I, J \subseteq \CC[x_0, \dots, x_n]$
@@ -288,7 +288,7 @@ \section{As ringed spaces}
 
 \begin{remark}
 	Unlike the case of \Cref{remark:meaning_name_coordinate_ring}, an element
-	of $\CC[V]$ no longer correspond to a function from $V$ to $\CC$;
+	of $\CC[V]$ no longer corresponds to a function from $V$ to $\CC$;
 	nevertheless, it is a function from $\VV(I) \subseteq \Aff^{n+1}$ to $\CC$.
 \end{remark}
 

tex/alg-geom/quasi-proj.tex

@@ -118,7 +118,7 @@ \section{Defining morphisms of baby ringed spaces}
 	The $f$-inverse image is $X \setminus \{\pm5\}$.
 	Thus the pullback is
 	\begin{align*}
-		f^\sharp\varphi \colon X \setminus \{\pm5\} &\to Y \setminus \{25\} \\
+		f^\sharp\varphi \colon X \setminus \{\pm5\} &\to \CC \\
 		\quad\text{by}\quad x &\mapsto \frac{1}{x^2-25}
 	\end{align*}
 	which is regular on $X \setminus \{\pm5\}$.
@@ -138,7 +138,7 @@ \section{Classifying the simplest examples}
 	\ii classify all the morphisms from $\Aff^m \to \Aff^n$, and
 	\ii classify all the morphisms from $\CP^m \to \CP^n$.
 \end{itemize}
-It what follows I will wave my hands a lot in claiming
+In what follows I will wave my hands a lot in claiming
 that something is a morphism, since doing so is mostly detail checking.
 The theorems which follow will give us alternative definitions
 of morphism which are more coordinate-based
@@ -175,10 +175,10 @@ \subsection{Affine classification}
 	First, remark that $f\pre(Y) = X$.
 	Now consider the regular function $\pi_1 \in \OO_Y(Y)$,
 	given by the projection $(y_1, \dots, y_n) \mapsto y_1$.
-	Thus we need $f \circ \pi_1$ to be regular on $X$.
+	Thus we need $\pi_1 \circ f$ to be regular on $X$.
 
 	But for affine varieties $\OO_X(X)$ is just the coordinate ring $\CC[X]$
-	and so we know there is a polynomial $P_1$ such that $f \circ \pi_1 = P_1$.
+	and so we know there is a polynomial $P_1$ such that $\pi_1 \circ f = P_1$.
 	Similarly for the other coordinates.
 \end{proof}
 
@@ -239,7 +239,7 @@ \subsection{Projective classification}
 
 \section{Some more applications and examples}
 \prototype{$\Aff^1 \injto \CP^1$ is a good one.}
-The previous section complete settles affine varieties to affine varieties,
+The previous section completely settles affine varieties to affine varieties,
 and projective varieties to projective varieties.
 However, the definition we gave at the start of the chapter
 works for \emph{any} baby ringed spaces,
@@ -340,7 +340,7 @@ \section{The hyperbola effect}
 Surprisingly, this doesn't work.
 \begin{example}
 	[Crucial example: punctured line is isomorphic to hyperbola]
-	Let $X = \Aff^1 \setminus \{0\}$ be an quasi-projective variety.
+	Let $X = \Aff^1 \setminus \{0\}$ be a quasi-projective variety.
 	We claim that in fact we have an isomorphism
 	\[ X \cong V = \VV(xy-1) \subseteq \Aff^2 \]
 	which shows that $X$ is still isomorphic to an affine variety.
@@ -429,8 +429,8 @@ \section{The hyperbola effect}
 \end{dproblem}
 
 \begin{problem}
-	Let $V$ be a affine variety
-	and let $W$ be a irreducible projective variety.
+	Let $V$ be an affine variety
+	and let $W$ be an irreducible projective variety.
 	Prove that $V \cong W$ if and only
 	if $V$ and $W$ are a single point.
 	\begin{hint}

tex/alg-geom/zariski.tex

@@ -135,7 +135,7 @@ \section{The Zariski topology on $\Aff^n$}
 \begin{moral}
 	The nonempty Zariski open sets are \emph{huge}.
 \end{moral}
-This is an important difference than what you're used to in topology.
+This is an important difference from what you're used to in topology.
 To be very clear:
 \begin{itemize}
 	\ii In the past, if I said something like
@@ -147,7 +147,7 @@ \section{The Zariski topology on $\Aff^n$}
 	should be thought of as saying ``is true for virtually all points,
 	other than those on certain curves''.
 \end{itemize}
-Indeed, ``open neighborhood'' is no longer really a accurate description.
+Indeed, ``open neighborhood'' is no longer really an accurate description.
 Nonetheless, in many pictures to follow,
 it will still be helpful to draw open neighborhoods as circles.
 
@@ -275,8 +275,8 @@ \section{Coordinate rings}
 \begin{remark}
 	[Meaning of the name ``coordinate ring'']
 	\label{remark:meaning_name_coordinate_ring}
-	We call the functions $x$, $y$ and $z$ above as the \vocab{coordinate
-	functions}, as they maps each point in the variety $V$ to its coordinate.
+	We call the functions $x$, $y$ and $z$ above the \vocab{coordinate
+	functions}, as they map each point in the variety $V$ to its coordinate.
 	So, the coordinate ring $\CC[V]$ is simply the ring generated by $\CC$ and
 	the coordinate functions.
 \end{remark}