path: root/controle-20240410.tex

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\begin{document}
\ifcorrige
\title{ACCQ205\\Final exam — answer key\\{\normalsize Algebraic curves}}
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\title{ACCQ205\\Final exam\\{\normalsize Algebraic curves}}
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\author{}
\date{2024-04-10}
\maketitle

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\noindent\textbf{Instructions.}

\textcolor{red}{xxx}

\medbreak

Answers can be written in English or French.

\medbreak

Use of written documents of any kind (such as handwritten or printed
notes, exercise sheets or books) is authorized.

Use of electronic devices of any kind is prohibited.

\medbreak

Duration: 2 hours

\ifcorrige
This answer key has \textcolor{red}{xxx} pages (cover page included).
\else
This exam has \textcolor{red}{xxx} pages (cover page included).
\fi

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\exercise

We say that a set of eight distinct points $p_0,\ldots,p_7$ in the
projective plane $\mathbb{P}^2$ over a field $k$ is a
\textbf{Möbius-Kantor configuration} when the points $p_0,p_1,p_3$ are
aligned, as well as $p_1,p_2,p_4$ and $p_2,p_3,p_5$ and so on
cyclically mod $8$, and no other set of three of the $p_i$ is aligned.
In other words, this means that $p_i,p_j,p_k$ are aligned if and only
if $\{i,j,k\} = \{\ell,\; \ell+1,\; \ell+3\}$ for some $\ell \in
\mathbb{Z}/8\mathbb{Z}$, where the subscripts are understood to be
mod $8$.

The following figure (which is meant as a \emph{symbolic
representation} of the configuration and not as an actual geometric
figure!) illustrates the setup and can help keep track of which points
are aligned with which:

\begin{center}
\vskip-7ex\leavevmode
\begin{tikzpicture}
\coordinate (P0) at (2cm,0);
\coordinate (P1) at (1.414cm,1.414cm);
\coordinate (P2) at (0,2cm);
\coordinate (P3) at (-1.414cm,1.414cm);
\coordinate (P4) at (-2cm,0);
\coordinate (P5) at (-1.414cm,-1.414cm);
\coordinate (P6) at (0,-2cm);
\coordinate (P7) at (1.414cm,-1.414cm);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P0) -- (P1) .. controls ($2.5*(P1)-1.5*(P0)$) and ($2.5*(P2)-1.5*(P1)$) .. (P3);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P1) -- (P2) .. controls ($2.5*(P2)-1.5*(P1)$) and ($2.5*(P3)-1.5*(P2)$) .. (P4);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P2) -- (P3) .. controls ($2.5*(P3)-1.5*(P2)$) and ($2.5*(P4)-1.5*(P3)$) .. (P5);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P3) -- (P4) .. controls ($2.5*(P4)-1.5*(P3)$) and ($2.5*(P5)-1.5*(P4)$) .. (P6);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P4) -- (P5) .. controls ($2.5*(P5)-1.5*(P4)$) and ($2.5*(P6)-1.5*(P5)$) .. (P7);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P5) -- (P6) .. controls ($2.5*(P6)-1.5*(P5)$) and ($2.5*(P7)-1.5*(P6)$) .. (P0);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P6) -- (P7) .. controls ($2.5*(P7)-1.5*(P6)$) and ($2.5*(P0)-1.5*(P7)$) .. (P1);
\draw[shorten <=-0.5cm, shorten >=-0.3cm] (P7) -- (P0) .. controls ($2.5*(P0)-1.5*(P7)$) and ($2.5*(P1)-1.5*(P0)$) .. (P2);
\fill[black] (P0) circle (2.5pt);
\fill[black] (P1) circle (2.5pt);
\fill[black] (P2) circle (2.5pt);
\fill[black] (P3) circle (2.5pt);
\fill[black] (P4) circle (2.5pt);
\fill[black] (P5) circle (2.5pt);
\fill[black] (P6) circle (2.5pt);
\fill[black] (P7) circle (2.5pt);
\node[anchor=west] at (P0) {$p_0$};
\node[anchor=south west] at (P1) {$p_1$};
\node[anchor=south] at (P2) {$p_2$};
\node[anchor=south east] at (P3) {$p_3$};
\node[anchor=east] at (P4) {$p_4$};
\node[anchor=north east] at (P5) {$p_5$};
\node[anchor=north] at (P6) {$p_6$};
\node[anchor=north west] at (P7) {$p_7$};
\end{tikzpicture}
\vskip-7ex\leavevmode
\end{center}

The goal of this exercise is to decide 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
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!)

\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
the questions will serve to compute the coordinates of the points.}
We denote $\ell_{ijk}$ the line through $p_i,p_j,p_k$ when it exists.

\textbf{(1)} Explain why we can assume, without loss of generality,
that $p_0=(1{:}0{:}0)$ and $p_1=(0{:}1{:}0)$ and $p_2=(0{:}0{:}1)$ and
$p_5=(1{:}1{:}1)$.  \emph{We shall henceforth do so.}

\textbf{(2)} Compute the coordinates of the lines $\ell_{013}$,
$\ell_{124}$, $\ell_{235}$, $\ell_{560}$ and $\ell_{702}$, and of the
point $p_3$.

\textbf{(3)} Explain why we can write, without loss of generality, the
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{(5)} Write the coordinates of the last remaining point $p_7$
in two different ways (using two different pairs of lines) and
conclude that $\xi$ must satisfy $1-\xi+\xi^2 = 0$.

\textbf{(6)} Deduce from questions (1)–(5) above that, if a
Möbius-Kantor configuration over $k$ exists, then there is $\xi\in k$
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.)

\textbf{(8)} Give two different examples of fields $k$, one infinite
and one finite, over which a Möbius-Kantor configuration exists, and
similarly two examples over which it does not exist.


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\exercise

The focus of this exercise is \textbf{Klein's quartic}, namely the
projective algebraic variety $C$ defined by the equation
\[
x^3 y + y^3 z + z^3 x = 0
\]
in $\mathbb{P}^2$ with coordinates $(x{:}y{:}z)$.  Note the symmetry
of this equation under cyclic permutation of the
coordinates\footnote{To dispel any possible confusion, this means
simultaneously replacing $x$ by $y$, $y$ by $z$ and $z$ by $x$.},
which will come in handy to simplify some computations.  To refer to
it more easily, we shall denote $f := x^3 y + y^3 z + z^3 x$ the
polynomial defining the equation of $C$.

We shall work over a field $k$ having characteristic $\not\in\{2,7\}$.
For simplicity, we shall also assume $k$ to be algebraically closed
(even though this won't matter at all).

\textbf{(1)} The following relation holds (this is a straightforward
computation, and it is not required to check it):
\[
-27xyz\,\frac{\partial f}{\partial x}
+(28x^3-3y^2 z)\,\frac{\partial f}{\partial y}
-9yz^2\,\frac{\partial f}{\partial z}
= 28x^6
\tag{$*$}
\]
What does the relation ($*$), together with the other two obtained by
cyclically permuting coordinates, tell us about the ideal generated by
$\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ and
$\frac{\partial f}{\partial z}$ in $k[x,y,z]$?  What does this imply
on the set of points where $\frac{\partial f}{\partial x}$,
$\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$
all vanish?

\smallskip

\emph{The previous question implies that $C$ is a (plane) curve.  The
following picture is a rough sketch of an affine part of $C$ over the
real field.}

\begin{center}
\begin{tikzpicture}
\begin{scope}[thick]
\clip (-3,-3) -- (3,-3) -- (3,3) -- (-3,3) -- cycle;
\draw (-3.000,5.251) .. controls (-2.667,4.397) and (-2.333,3.618) .. (-2.000,2.946) ;
\draw (-2.000,2.946) .. controls (-1.833,2.610) and (-1.667,2.301) .. (-1.500,2.028);
\draw (-1.500,2.028) .. controls (-1.333,1.755) and (-1.167,1.519) .. (-1.000,1.325) ;
\draw (-1.000,1.325) .. controls (-0.833,1.130) and (-0.667,0.981) .. (-0.500,0.846) ;
\draw (-0.500,0.846) .. controls (-0.417,0.779) and (-0.333,0.716) .. (-0.250,0.638) ;
\draw (-0.250,0.638) .. controls (-0.208,0.600) and (-0.167,0.558) .. (-0.125,0.501) ;
\draw (-0.125,0.501) .. controls (-0.104,0.473) and (-0.083,0.441) .. (-0.062,0.397) ;
\draw (-0.062,0.397) .. controls (0,0.265) and (0,0.133) .. (0,0) ;
\draw (0,0) .. controls (0,-0.133) and (0,-0.265) .. (0.062,-0.397) ;
\draw (0.062,-0.397) .. controls (0.083,-0.441) and (0.104,-0.471) .. (0.125,-0.499) ;
\draw (0.125,-0.499) .. controls (0.167,-0.553) and (0.208,-0.590) .. (0.250,-0.622) ;
\draw (0.250,-0.622) .. controls (0.333,-0.684) and (0.417,-0.720) .. (0.500,-0.741) ;
\draw (0.500,-0.741) .. controls (0.667,-0.783) and (0.833,-0.755) .. (1.000,-0.682) ;
\draw (1.000,-0.682) .. controls (1.167,-0.610) and (1.333,-0.501) .. (1.500,-0.422) ;
\draw (1.500,-0.422) .. controls (1.667,-0.343) and (1.833,-0.288) .. (2.000,-0.248) ;
\draw (2.000,-0.248) .. controls (2.333,-0.168) and (2.667,-0.136) .. (3.000,-0.111) ;
\draw (-3.000,-5.140) .. controls (-2.667,-4.261) and (-2.333,-3.452) .. (-2.000,-2.694) ;
\draw (-2.000,-2.694) .. controls (-1.833,-2.315) and (-1.667,-1.962) .. (-1.500,-1.552) ;
\draw (-1.500,-1.552) .. controls (-1.458,-1.449) and (-1.417,-1.346) .. (-1.375,-1.209) ;
\draw (-1.375,-1.209) .. controls (-1.315,-1.013) and (-1.263,-0.817) .. (-1.375,-0.621) ;
\draw (-1.375,-0.621) .. controls (-1.417,-0.548) and (-1.458,-0.511) .. (-1.500,-0.477) ;
\draw (-1.500,-0.477) .. controls (-1.667,-0.339) and (-1.833,-0.295) .. (-2.000,-0.252) ;
\draw (-2.000,-0.252) .. controls (-2.333,-0.166) and (-2.667,-0.136) .. (-3.000,-0.111) ;
\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) {$\scriptstyle y/z =: v$};
\end{tikzpicture}
\end{center}

We now define the three points $a := (1{:}0{:}0)$, $b := (0{:}1{:}0)$
and $c := (0{:}0{:}1)$ (which obviously lie on $C$).

\textbf{(2)} List all points of $C$ where $x$ vanishes.  Do the same
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
the picture?  What is the tangent line at the point $c$?  What about
$a$ and $b$?

\textbf{(4)} Considering $v := \frac{y}{z}$ as a rational function
on $C$, explain why it vanishes at order exactly $1$ at $c$, that
is\footnote{We write $\ord_p(h)$ for the order at a point $p \in C$ of
a rational function $h \in k(C)$.  By the way, please note that
$x,y,z$ themselves do not belong to $k(C)$ (they are not functions and
have no value by themselves), so we cannot speak of $\ord_p(x)$.},
$\ord_c(v) = 1$.  Explain why $\ord_c(u) = \ord_c(u^3 v + v^3)$ where
$u := \frac{x}{z}$ and deduce that $\ord_c(u) = 3$.  Deduce the order
at $c$ of $\frac{y}{x}$ (which is also $\frac{v}{u}$).

\textbf{(5)} By using symmetry, compute the order at each one of the
three points $a,b,c$ of each one of the three functions $\frac{x}{z}$,
$\frac{y}{x}$ and $\frac{z}{y}$.  Explain why there are no points
(of $C$) other than $a,b,c$ where any of these functions (on $C$)
vanishes or has a pole.  Summarize this by writing the principal
divisors $\divis(\frac{x}{z})$, $\divis(\frac{y}{x})$ and
$\divis(\frac{z}{y})$ associated with these three functions.


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\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, $n = pq+p+q$).

Assume $k$ is a field.  To simplify presentation, assume $k$ is
algebraically closed (even though this won't matter at all).

Given $p,q\in\mathbb{N}$, the Segre embedding of $\mathbb{P}^p \times
\mathbb{P}^q$ is the map $\psi$ given by:
\[
\begin{aligned}
\psi\colon & \mathbb{P}^p \times \mathbb{P}^q \to \mathbb{P}^n\\
&((x_0:\cdots:x_p),\, (y_0:\cdots:y_q)) \mapsto (x_0 y_0 : x_0 y_1 : \cdots
: x_0 y_q : x_1 y_0 : \cdots : x_p y_q)\\
\end{aligned}
\]
where $n = (p+1)(q+1)-1$ and the coordinates of the endpoint consist
of every product $x_i y_j$ with $0\leq i\leq p$ and $0\leq j\leq q$
(in some order which doesn't really matter: here we have chosen the
lexicographic ordering).

Note that with the definitions given in this course, we cannot state
that $\psi$ is a morphism of algebraic varieties (although it
certainly \emph{should} be one), because we did not define a “product
variety”\footnote{In fact, the Segre embedding is one way of doing
this.} $\mathbb{P}^p \times \mathbb{P}^q$.  But we can still consider
it as a function.

Let us label $(z_{0,0} : z_{0,1} : \cdots : z_{p,q})$ the homogeneous
coordinates in $\mathbb{P}^n$ (that is, $z_{i,j}$ with $0\leq i\leq p$
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 $0\leq
j,j'\leq q$.

\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.

\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
equation.  Taking $z_{0,0}=0$ as the plane at infinity in
$\mathbb{P}^3$, give the equation of the affine part $S \cap
\mathbb{A}^3$.  Similarly taking $x_0=0$ (resp. $y_0=0$) as the point
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 $S$.

\textbf{(4)} Conversely, explain why given a 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
$\psi$ is a bijection between $\mathbb{P}^p \times \mathbb{P}^q$
and $S$.

\textbf{(5)} Call $\pi\colon S\to \mathbb{P}^p\times\mathbb{P}^q$ the
inverse bijection of $\psi$, and call $\pi',\pi''$ its two components.
(In other words, if $s = (z_{0,0} : \cdots : z_{p,q})$ is in $S$ then
$\pi'(s) = (x_0:\cdots:x_p) \in \mathbb{P}^p$ 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.)



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