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| -rw-r--r-- | controle-20230412.tex | 34 |
1 files changed, 17 insertions, 17 deletions
diff --git a/controle-20230412.tex b/controle-20230412.tex index 9332301..6cb3728 100644 --- a/controle-20230412.tex +++ b/controle-20230412.tex @@ -136,11 +136,11 @@ denote as $[u_0{:}u_1{:}u_2{:}u_3]$ (a point of the “dual” $\mathbb{P}^3$). Our goal is to find a representation for lines. -It may be convenient, if so desired, to call $\langle w\rangle$ the -point in projective space $\mathbb{P}^{m-1}(k)$ defined by a vector -$w\neq 0$ in $k^m$ (i.e., if $w = (w_0,\ldots,w_m)$ then $\langle w -\rangle = (w_0{:}\cdots{:}w_m)$), that is, the class of $w$ under -collinearity. +{\footnotesize It may be convenient, if so desired, to call $\langle + w\rangle$ the point in projective space $\mathbb{P}^{m-1}(k)$ + defined by a vector $w\neq 0$ in $k^m$ (i.e., if $w = + (w_0,\ldots,w_m)$ then $\langle w \rangle = (w_0{:}\cdots{:}w_m)$), + that is, the class of $w$ under collinearity. \par} \bigskip @@ -181,10 +181,10 @@ x,y\in V\}$ is the line spanned by $u\wedge v$. \textbf{(3)} Deduce from (2) that if $L \subseteq \mathbb{P}^3(k)$ is a line, then $(w_{0,1}{:}w_{0,2}{:}w_{0,3} {:} \penalty0 w_{1,2}{:}w_{1,3}{:}w_{2,3})$, where $w_{i,j} := x_i y_j - x_j y_i$ as -above, and $(x_0{:}x_1{:}x_2{:}x_3)$ and $(y_0{:}y_1{:}y_2{:}y_3)$ are -two distinct points on $L$, is a well-defined point in -$\mathbb{P}^5(k)$, not depending on the chosen points on $L$ nor on -the homogeneous coordinates representing them. +above, and where $(x_0{:}x_1{:}x_2{:}x_3)$ and +$(y_0{:}y_1{:}y_2{:}y_3)$ are two distinct points on $L$, is a +well-defined point in $\mathbb{P}^5(k)$, not depending on the chosen +points on $L$ nor on the homogeneous coordinates representing them. \begin{answer} Calling $\langle w\rangle$ the point in projective space @@ -256,12 +256,12 @@ w_{1,2}{:}w_{1,3}{:}w_{2,3})$ in $\mathbb{P}^5(k)$ satisfies ($\dagger$) (viꝫ. belongs to the Plücker quadric), and assuming also that $w_{0,3} \neq 0$, show that the two points $(w_{0,3}{:}w_{1,3}{:}w_{2,3}{:}0)$ and -$(0{:}w_{0,1}{:}w_{0,2}{:}w_{0,3})$ are meaningful and distinct, and -that the line joining them has the Plücker coordinates -$(w_{0,1}:\cdots:w_{2,3})$ that were given. (\emph{Hint:} -\underline{first} compute $(w_{0,3},w_{1,3},w_{2,3},0) \wedge -(0,w_{0,1},w_{0,2},w_{0,3})$ and then use the result, with the Plücker -relation and the fact that $w_{0,3} \neq 0$ to conclude.) +$(0{:}w_{0,1}{:}w_{0,2}{:}w_{0,3})$ in $\mathbb{P}^3(k)$ are +meaningful and distinct, and that the line joining them has the +Plücker coordinates $(w_{0,1}:\cdots:w_{2,3})$ that were given. +(\emph{Hint:} \underline{first} compute $(w_{0,3},w_{1,3},w_{2,3},0) +\wedge (0,w_{0,1},w_{0,2},w_{0,3})$ and then use the result, with the +Plücker relation and the fact that $w_{0,3} \neq 0$ to conclude.) \begin{answer} We straightforwardly compute $(w_{0,3},w_{1,3},w_{2,3},0) \wedge @@ -310,12 +310,12 @@ lines $L$ in $\mathbb{P}^3(k)$ and the set of $k$-points in the Plücker quadric defined by ($\dagger$) in $\mathbb{P}^5$; we know how to compute Plücker coordinates from two distinct points lying on $L$ (by definition). We now wish to compute Plücker coordinates for a -line defined as the the intersection of two planes. +line that is described as the the intersection of two planes. \textbf{(8)} Rephrase (4) to deduce that, if $L$ is a line with Plücker coordinates $(w_{0,1}:\cdots:w_{2,3})$, then the planes $[w_{1,2} : {-w_{0,2}} : w_{0,1} : 0]$ and $[0 : w_{2,3} : {-w_{1,3}} - : w_{1,2}]$ contain $L$. Now consider these as points in the + : w_{1,2}]$ both contain $L$. Now consider these as points in the dual $\mathbb{P}^3$ and show that the Plücker coordinates of the line $L^*$ joining the two points in question are: $[w_{2,3} : {-w_{1,3}} : w_{1,2} : w_{0,3} : {-w_{0,2}} : w_{0,1}]$, provided $w_{1,2} \neq |