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100644 --- a/Geometria 1/Spazi affini/2023-05-05, Affinità e spazio proiettivo/main.tex +++ b/Geometria 1/Spazi affini/2023-05-05, Affinità e spazio proiettivo/main.tex @@ -401,4 +401,49 @@ modificata, per $D \cap D' \neq \emptyset$) utilizzando questa osservazione, così come si sarebbe potuto dimostrare che $(D + D')_0 = D_0 + D_0'$. \end{remark} + + \hr + + \begin{remark} [punti fissi di un'applicazione affine] + Si consideri un'applicazione affine $f$ di $\AnK$. Allora esistono $A \in M(n, \KK) \setminus \{0\}$ e $\vec b \in \AnK$ tali + per cui $f(\x) = A \x + \vec b$ $\forall \x \in \AnK$. In particolare, $f$ ammette punti fissi se + esiste $\x \in \AnK \mid f(\x) = \x \iff A \x + \vec b = \x \iff (A-I) \x = -\vec b \iff \vec b \in \Im(A-I)$. \\ + + Ciò è sicuramente vero se $A$ non ammette $1$ come autovalore (infatti in tal caso $A-I$ è invertibile, e quindi + in particolare è surgettiva). + \end{remark} + + \begin{example} + Si consideri $f \in A(\Aa_1(\KK))$. Allora esistono $a$, $b \in \Aa_1(\KK)$ tali per cui $f(x) = ax+b$ $\forall x \in \Aa_1(\KK)$. \\ + + Se $a \neq 1$ (ossia se non ammette $1$ come autovalore), $f$ ammette un punto fisso, ossia + $x = -\frac{b}{a-1}$. Altrimenti, se $a = 1$ e $b \neq 0$, $f$ è una traslazione (e quindi non ammette punti fissi). + Allora, indicando con $\Fix(f) = \{ x \in \Aa_1(\KK) \mid f(x) = x \}$ l'insieme dei punti fissi + di $f$, vale sicuramente che $\abs{\Fix(f)} \leq 1$, escludendo il caso in cui $a = 1$ e $b = 0$. \\ + + Inoltre, vale che $A(\Aa_1(\KK))$ agisce transitivamente su $\Aa_1(\KK)$, ossia esiste sempre un'applicazione + affine tale per cui $x \mapsto y$, dati $x$, $y \in \Aa_1(\KK)$. Ciononostante $A(\Aa_1(\KK))$ non agisce + liberamente su $\Aa_1(\KK)$, ed in particolare vale che $\Stab(x) \cong \GL(1, \KK)$\footnote{È sufficiente + mappare ogni affinità alla propria matrice $A$, dal momento che $\vec b$ è già univocamente determinante.}. + Un risultato analogo vale anche per gli altri spazi affini: per esempio vale ancora che + $\Stab(\x) \cong \GL(2, \KK)$ $\forall \x \in \Aa_2(\KK)$, sull'azione generata da $A(\Aa_2(\KK))$ su $\Aa_2(\KK)$. \\ + + Infine, date due coppie di punti $(P, P')$ e $(Q, Q')$ con $P$, $P'$, $Q$, $Q' \in \Aa_1(\KK)$, + $P \neq P'$ e $Q \neq Q'$, $A(\Aa_1(\KK))$ agisce in maniera semplicemente transitiva mappando una coppia + di punti all'altra\footnote{Infatti l'unica applicazione affine che manda una coppia di punti nella stessa coppia di punti è obbligatoriamente l'identità, come visto sopra nello studio di $\Fix(f)$. Infine si osserva che esiste sempre un'applicazione che manda una coppia di punti nell'altra.}. + \end{example} + + \begin{remark} [rapporto semplice] Siano $P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$ e $Q_3$ punti di $\Aa_1(\KK)$ con + $P_1$, $P_2$, $P_3$ distinti e $Q_1$, $Q_2$, $Q_3$ distinti. Allora esiste un'unica applicazione affine $f$ + tale per cui $f(P_1) = Q_1$ e $f(P_2) = Q_2$, dacché $P_1$ e $P_2$ formano un riferimento affine di $\Aa_1(\KK)$. \\ + + In particolare $f$ è tale che $f(P_3) = Q_3$ se e solo se, detto $\lambda \in \KK$ il parametro tale per + cui $P_3 = (1-\lambda) P_1 + \lambda P_2$, $Q_3 = f(P_3) = (1-\lambda) f(P_1) + \lambda f(P_2) = + (1-\lambda) Q_1 + \lambda Q_2$, ossia se e solo se $\lambda$ è lo stesso parametro con cui si scrive $Q_2$ rispetto + al riferimento affine dato da $Q_1$ e $Q_2$. In particolare ciò è vero se e solo se vale che $\frac{P_3 - P_1}{P_2 - P_1} = \lambda = \frac{Q_3 - Q_1}{Q_2 - Q_1}$. \\ + + Il rapporto $\frac{P_3 - P_1}{P_2 - P_1}$ si dice \textbf{rapporto semplice} della terna di punti $P_1$, $P_2$ e $P_3$. + Si conclude dunque che tale $f$ esiste (ed è unica) se e solo se i rapporti semplici delle due terne di punti + coincidono. + \end{remark} \end{document}