From 79c2ef3456b77e5b3fdcc2ab6d26fd052f6a053d Mon Sep 17 00:00:00 2001
From: "David A. Madore" <david+git@madore.org>
Date: Mon, 8 Apr 2024 15:15:52 +0200
Subject: Re-read test.

---
 controle-20240410.tex | 74 ++++++++++++++++++++++++++-------------------------
 1 file changed, 38 insertions(+), 36 deletions(-)

(limited to 'controle-20240410.tex')

diff --git a/controle-20240410.tex b/controle-20240410.tex
index 9e6ad1d..d3e54e3 100644
--- a/controle-20240410.tex
+++ b/controle-20240410.tex
@@ -31,6 +31,7 @@
 \newcommand\exercise{%
 \refstepcounter{comcnt}\bigskip\noindent\textbf{Exercise~\thecomcnt.}\par\nobreak}
 \renewcommand{\qedsymbol}{\smiley}
+\renewcommand{\thefootnote}{\fnsymbol{footnote}}
 %
 \newcommand{\id}{\operatorname{id}}
 \newcommand{\alg}{\operatorname{alg}}
@@ -173,25 +174,24 @@ are aligned with which:
 \vskip-7ex\leavevmode
 \end{center}
 
-The goal of this exercise is to decide over which fields $k$ a
+The goal of this exercise is to determine over which fields $k$ a
 Möbius-Kantor configuration exists, and compute the coordinates of its
 points.
 
-We fix a field $k$ and the word “point”, in what follows, will refer
+We fix a field $k$.  The word “point”, in what follows, will refer
 to an element of $\mathbb{P}^2(k)$, in other words, a point with
 coordinates in $k$ (that is, a $k$-point).
 
-We shall write as $(x{:}y{:}z)$ the coordinates of a point in
-$\mathbb{P}^2(k)$, and as $[u{:}v{:}w]$ the line $\{ux+vy+wz = 0\}$.
-Recall that the line through $(x_1{:}y_1{:}z_1)$ and
-$(x_2{:}y_2{:}z_2)$ (assumed distinct) is given by the formula $[(y_1
-  z_2 - y_2 z_1) : (z_1 x_2 - z_2 x_1) : (x_1 y_2 - x_2 y_1)]$, and
-that the same formula (exchanging parentheses and square brackets) can
-also be used to compute the intersection of two distinct lines.  (This
-may not always be the best or simplest way\footnote{For example, you
-shouldn't need this formula to notice that the line through
-$(42{:}0{:}0)$ and $(0{:}1729{:}0)$ is $[0{:}0{:}1]$.} to compute
-coordinates, however!)
+We shall write as $(x{:}y{:}z)$ the coordinates of a point, and as
+$[u{:}v{:}w]$ the line $\{ux+vy+wz = 0\}$.  Recall that the line
+through $(x_1{:}y_1{:}z_1)$ and $(x_2{:}y_2{:}z_2)$ (assumed distinct)
+is given by the formula $[(y_1 z_2 - y_2 z_1) : (z_1 x_2 - z_2 x_1) :
+  (x_1 y_2 - x_2 y_1)]$, and that the same formula (exchanging
+parentheses and square brackets) can also be used to compute the
+intersection of two distinct lines.  (This may not always be the best
+or simplest way\footnote{For example, one shouldn't need this formula
+  to notice that the line through $(42{:}0{:}0)$ and $(0{:}1729{:}0)$
+  is $[0{:}0{:}1]$.} to compute coordinates, however!)
 
 \emph{We assume for questions (1)–(5) below that $p_0,\ldots,p_7$ is a
 Möbius-Kantor configuration of points (over the given field $k$), and
@@ -211,8 +211,8 @@ coordinates of $p_4$ in the form $(0{:}\xi{:}1)$ for some $\xi$
 (in $k$).  (Note that two things need to be explained here: why the
 first coordinate is $0$ and why the last can be taken to be $1$.)
 
-\textbf{(4)} Now compute the coordinates of the lines $\ell_{346}$ and
-$\ell_{457}$, of the point $p_6$, and of the line $\ell_{671}$.
+\textbf{(4)} Now compute the coordinates of the line $\ell_{346}$, of
+the point $p_6$, and of the lines $\ell_{457}$ and $\ell_{671}$.
 
 \textbf{(5)} Write the coordinates of the last remaining point $p_7$
 in two different ways (using two different pairs of lines) and
@@ -225,8 +225,8 @@ such that $1-\xi+\xi^2 = 0$.
 \textbf{(7)} Conversely, using the coordinate computations performed
 in questions (2)–(5), explain why, if there is $\xi\in k$ such that
 $1-\xi+\xi^2 = 0$, then a Möbius-Kantor configuration over $k$ exists.
-(A long explanation is not required, but you should at least explain
-what checks need be done.)
+(A long explanation is not required, but at least explain what checks
+need be done.)
 
 \textbf{(8)} Give two different examples of fields $k$, one infinite
 and one finite, over which a Möbius-Kantor configuration exists, and
@@ -310,8 +310,8 @@ real field.}
 \end{scope}
 \draw[->, shorten <=-0.1cm, shorten >=-0.1cm, thin] (-3,0) -- (3,0);
 \draw[->, shorten <=-0.1cm, shorten >=-0.1cm, thin] (0,-3) -- (0,3);
-\node[anchor=west] at (3,0) {$\scriptstyle x/z =: u$};
-\node[anchor=south] at (0,3) {$\scriptstyle y/z =: v$};
+\node[anchor=west] at (3,0) {$\scriptstyle x/z \,=:\, u$};
+\node[anchor=south] at (0,3) {$\scriptstyle y/z \,=:\, v$};
 \end{tikzpicture}
 \end{center}
 
@@ -323,7 +323,7 @@ for $y$ and $z$.
 
 \textbf{(3)} Where do the points $a,b,c$ lie on the printed picture?
 (If they do not lie on the picture, show the direction in which they
-should be.)  What is the equation of the affine part of $C$ drawn on
+would be.)  What is the equation of the affine part of $C$ drawn on
 the picture?  What is the tangent line at the point $c$?  What about
 $a$ and $b$?
 
@@ -353,10 +353,11 @@ $\divis(\frac{z}{y})$ associated with these three functions.
 
 \exercise
 
-This exercise is about the \textbf{Segre embedding}, which is a way to
-map the product $\mathbb{P}^p \times \mathbb{P}^q$ of two projective
-spaces to a larger projective space $\mathbb{P}^n$ (with, as we shall
-see, $n = pq+p+q$).
+This exercise is about the \textbf{Segre embedding}\footnote{French:
+  “plongement de Segre”}, which is a way to map the product
+$\mathbb{P}^p \times \mathbb{P}^q$ of two projective spaces to a
+larger projective space $\mathbb{P}^n$ (with, as we shall see, $n =
+pq+p+q$).
 
 Assume $k$ is a field.  To simplify presentation, assume $k$ is
 algebraically closed (even though this won't matter at all).
@@ -388,17 +389,17 @@ and $0\leq j\leq q$), so that $\psi$ is given simply by “$z_{i,j} =
 x_i y_j$”.
 
 We finally consider the Zariski closed subset $S$ of $\mathbb{P}^n$,
-known as the \textbf{Segre variety}, defined by the equations $z_{i,j}
-z_{i',j'} = z_{i,j'} z_{i',j}$ for all $0\leq i,i'\leq p$ and $0\leq
-j,j'\leq q$.
+known as the \textbf{Segre variety}, defined in $\mathbb{P}^n$ by the
+equations $z_{i,j} z_{i',j'} = z_{i,j'} z_{i',j}$ for all $0\leq
+i,i'\leq p$ and $0\leq j,j'\leq q$.
 
 \medskip
 
 \textbf{(1)} Explain why the map $\psi$ is well-defined, i.e., the
-definition above makes sense: carefully list the properties that need
-to be checked, and do so.  Explain why $S$ is indeed a Zariski closed
-subset of $\mathbb{P}^n$: again, carefully state what needs to be
-checked before doing so.
+definition given above makes sense: carefully list the properties that
+need to be checked, and do so.  Explain why $S$ is indeed a Zariski
+closed subset of $\mathbb{P}^n$: again, carefully state what needs to
+be checked before doing so.
 
 \textbf{(2)} Consider in this question the special case $p=q=1$ (so
 $n=3$).  Simplify the definition of $S$ in this case down to a single
@@ -409,9 +410,10 @@ at infinity in $\mathbb{P}^1$, describe $\psi$ on $\mathbb{A}^1 \times
 \mathbb{A}^1$.
 
 \textbf{(3)} Returning to the case of general $p$ and $q$, show that
-the image of $\psi$ is included in $S$.
+the image of $\psi$ is contained in $S$, that is, $\psi(\mathbb{P}^p
+\times \mathbb{P}^q) \subseteq S$.
 
-\textbf{(4)} Conversely, explain why given a point $(z_{0,0} : \cdots
+\textbf{(4)} Conversely, explain why for each point $(z_{0,0} : \cdots
 : z_{p,q})$ in $S$ there is a unique pair of points $((x_0 : \cdots :
 x_p), (y_0 : \cdots : y_q))$ in $\mathbb{P}^p \times \mathbb{P}^q$
 which maps to the given point under $\psi$: in other words, show that
@@ -425,9 +427,9 @@ $\pi'(s) = (x_0:\cdots:x_p) \in \mathbb{P}^p$ and $\pi''(s) =
 (y_0:\cdots:y_p) \in \mathbb{P}^q$ are the unique points such that
 $(\pi'(s),\pi''(s))$ maps to $s$ under $\psi$.)  Show that the maps
 $\pi' \colon S \to \mathbb{P}^p$ and $\pi'' \colon S \to \mathbb{P}^q$
-are morphisms of algebraic varieties.  (If you find this too
-difficult, consider the special case $p=q=1$, and at least try to
-explain what needs to be checked.)
+are morphisms of algebraic varieties.  (If this seems too difficult,
+consider the special case $p=q=1$, and at least try to explain what
+needs to be checked.)
 
 
 
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