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Diffstat (limited to 'controle-20230412.tex')
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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 $X$? + |