More re-reading and minor changes.

1 files changed, 17 insertions, 17 deletions

diff --git a/controle-20230412.tex b/controle-20230412.tex
index 9332301..6cb3728 100644
--- a/controle-20230412.tex
+++ b/controle-20230412.tex
@@ -136,11 +136,11 @@ denote as $[u_0{:}u_1{:}u_2{:}u_3]$ (a point of the
 “dual” $\mathbb{P}^3$).  Our goal is to find a representation for
 lines.
 
-It may be convenient, if so desired, to call $\langle w\rangle$ the
-point in projective space $\mathbb{P}^{m-1}(k)$ defined by a vector
-$w\neq 0$ in $k^m$ (i.e., if $w = (w_0,\ldots,w_m)$ then $\langle w
-\rangle = (w_0{:}\cdots{:}w_m)$), that is, the class of $w$ under
-collinearity.
+{\footnotesize It may be convenient, if so desired, to call $\langle
+  w\rangle$ the point in projective space $\mathbb{P}^{m-1}(k)$
+  defined by a vector $w\neq 0$ in $k^m$ (i.e., if $w =
+  (w_0,\ldots,w_m)$ then $\langle w \rangle = (w_0{:}\cdots{:}w_m)$),
+  that is, the class of $w$ under collinearity. \par}
 
 \bigskip
 
@@ -181,10 +181,10 @@ x,y\in V\}$ is the line spanned by $u\wedge v$.
 \textbf{(3)} Deduce from (2) that if $L \subseteq \mathbb{P}^3(k)$ is
 a line, then $(w_{0,1}{:}w_{0,2}{:}w_{0,3} {:} \penalty0
 w_{1,2}{:}w_{1,3}{:}w_{2,3})$, where $w_{i,j} := x_i y_j - x_j y_i$ as
-above, and $(x_0{:}x_1{:}x_2{:}x_3)$ and $(y_0{:}y_1{:}y_2{:}y_3)$ are
-two distinct points on $L$, is a well-defined point in
-$\mathbb{P}^5(k)$, not depending on the chosen points on $L$ nor on
-the homogeneous coordinates representing them.
+above, and where $(x_0{:}x_1{:}x_2{:}x_3)$ and
+$(y_0{:}y_1{:}y_2{:}y_3)$ are two distinct points on $L$, is a
+well-defined point in $\mathbb{P}^5(k)$, not depending on the chosen
+points on $L$ nor on the homogeneous coordinates representing them.
 
 \begin{answer}
 Calling $\langle w\rangle$ the point in projective space
@@ -256,12 +256,12 @@ w_{1,2}{:}w_{1,3}{:}w_{2,3})$ in $\mathbb{P}^5(k)$
 satisfies ($\dagger$) (viꝫ. belongs to the Plücker quadric), and
 assuming also that $w_{0,3} \neq 0$, show that the two points
 $(w_{0,3}{:}w_{1,3}{:}w_{2,3}{:}0)$ and
-$(0{:}w_{0,1}{:}w_{0,2}{:}w_{0,3})$ are meaningful and distinct, and
-that the line joining them has the Plücker coordinates
-$(w_{0,1}:\cdots:w_{2,3})$ that were given.  (\emph{Hint:}
-\underline{first} compute $(w_{0,3},w_{1,3},w_{2,3},0) \wedge
-(0,w_{0,1},w_{0,2},w_{0,3})$ and then use the result, with the Plücker
-relation and the fact that $w_{0,3} \neq 0$ to conclude.)
+$(0{:}w_{0,1}{:}w_{0,2}{:}w_{0,3})$ in $\mathbb{P}^3(k)$ are
+meaningful and distinct, and that the line joining them has the
+Plücker coordinates $(w_{0,1}:\cdots:w_{2,3})$ that were given.
+(\emph{Hint:} \underline{first} compute $(w_{0,3},w_{1,3},w_{2,3},0)
+\wedge (0,w_{0,1},w_{0,2},w_{0,3})$ and then use the result, with the
+Plücker relation and the fact that $w_{0,3} \neq 0$ to conclude.)
 
 \begin{answer}
 We straightforwardly compute $(w_{0,3},w_{1,3},w_{2,3},0) \wedge
@@ -310,12 +310,12 @@ lines $L$ in $\mathbb{P}^3(k)$ and the set of $k$-points in the
 Plücker quadric defined by ($\dagger$) in $\mathbb{P}^5$; we know how
 to compute Plücker coordinates from two distinct points lying on $L$
 (by definition).  We now wish to compute Plücker coordinates for a
-line defined as the the intersection of two planes.
+line that is described as the the intersection of two planes.
 
 \textbf{(8)} Rephrase (4) to deduce that, if $L$ is a line with
 Plücker coordinates $(w_{0,1}:\cdots:w_{2,3})$, then the planes
 $[w_{1,2} : {-w_{0,2}} : w_{0,1} : 0]$ and $[0 : w_{2,3} : {-w_{1,3}}
-  : w_{1,2}]$ contain $L$.  Now consider these as points in the
+  : w_{1,2}]$ both contain $L$.  Now consider these as points in the
 dual $\mathbb{P}^3$ and show that the Plücker coordinates of the line
 $L^*$ joining the two points in question are: $[w_{2,3} : {-w_{1,3}} :
   w_{1,2} : w_{0,3} : {-w_{0,2}} : w_{0,1}]$, provided $w_{1,2} \neq