Add one more (difficult) question.

1 files changed, 26 insertions, 1 deletions

diff --git a/controle-20230412.tex b/controle-20230412.tex
index 6cb3728..9a12ee8 100644
--- a/controle-20230412.tex
+++ b/controle-20230412.tex
@@ -103,7 +103,7 @@ Duration: 2 hours
 \ifcorrige
 This answer key has 6 pages (cover page included).
 \else
-This exam has 3 pages (cover page included).
+This exam has 4 pages (cover page included).
 \fi
 
 \vfill
@@ -566,6 +566,31 @@ Z_4-\sqrt{-1}\,Z_5$ is invertible, there are still the same number of
 points.
 \end{answer}
 
+\bigskip
+
+\centerline{\hbox to3truecm{\hrulefill}}
+
+\medskip
+
+(This question is more difficult; it is independent of (13)\&(14).)
+
+\textbf{(15)} Let $h \in k[t_0,t_1,t_2,t_3]$ be a homogeneous
+polynomial, so that it defines a Zariski closed set (hypersurface) $X
+:= \{h(x_0,x_1,x_2,x_3) = 0\}$ in $\mathbb{P}^3$.  Show that the of
+lines contained in $X$ defines a Zariski closed subset $Y$ of the
+Plücker quadric in $\mathbb{P}^5$.  (To be completely clear, this
+means\footnote{Here $k^{\alg}$ denotes the algebraic closure of $k$,
+  but feel free to assume that $k$ is algebraically closed ($k =
+  k^{\alg}$) in this question.}: there is a Zariski closed set $Y$ in
+$\mathbb{P}^5$, defined over $k$ and contained in the Plücker quadric
+$Q$ (defined by $\dagger$), such, for $w \in Q(k^{\alg})$, we have $w
+\in Y(k^{\alg})$ if and only if $L_w \subseteq X(k^{\alg})$, where
+$L_w$ denotes the line in $\mathbb{P}^3(k^{\alg})$ having Plücker
+coordinates $w$.)
+
+The important part of this question is: how can we compute equations
+for $Y$ given the equation $h=0$ of $X$?
+