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| -rw-r--r-- | controle-20240410.tex | 74 |
1 files changed, 38 insertions, 36 deletions
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.) |