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Given non-zero $v$ we want to describe local coordinates of $P(V)$ in a neighborhood of the projective point $[v]={\Bbb C}v$. We do so through the orthogonal complement: $T_v=\{z: \langle a,v \rangle=0\}$.

Except for projective points in $P(T_v)$ we will expres any $[x]$ in the form $[v+w]$ for a suitable $w\in T_v$. To get this map let $x\in V\setminus T_v$ and let us seek $\lambda$ and $w\in T_v$ for which $$ x = \lambda (v+w) $$ Taking scalar product with $v$ we get: $\langle x,v\rangle = \lambda \langle v,v\rangle$ from which we first isolate $\lambda$ and insert in the first to obtain $w$: $$ \phi_v(x):= w = \frac{\langle v,v\rangle}{\langle x,v\rangle}x-v. $$ Now, for any non-zero $\mu$ we have $\phi_v(\mu x)=\phi_v(x)$ so this lifts to a well-defined map: $\hat{\phi}_v([x]) = \phi_v(x)$ and $\Psi_v^{-1}=\hat{\phi}_v: P(V)\setminus P(T_v) \rightarrow T_v$ is the wanted coordinate map.

Exercise: Given $v,v'$ and corresponding $[x]\in A_v\cap A_{v'}$ find the change of coordinate map taking $w\in T_v \mapsto w'\in T_{v'}$

Given non-zero $v$ we want to describe local coordinates of $P(V)$ in a neighborhood of the projective point $[v]={\Bbb C}v$. We do so through the orthogonal complement: $T_v=\{z: \langle a,v \rangle=0\}$.

Except for projective points in $P(T_v)$ we will expres any $[x]$ in the form $[v+w]$ for a suitable $w\in T_v$. To get this map let $x\in V\setminus T_v$ and let us seek $\lambda$ and $w\in T_v$ for which $$ x = \lambda (v+w) $$ Taking scalar product with $v$ we get: $\langle x,v\rangle = \lambda \langle v,v\rangle$ from which we first isolate $\lambda$ and insert in the first to obtain $w$: $$ \phi_v(x):= w = \frac{\langle v,v\rangle}{\langle x,v\rangle}x-v. $$ Now, for any non-zero $\mu$ we have $\phi_v(\mu x)=\phi_v(x)$ so this lifts to a well-defined map: $\hat{\phi}_v([x]) = \phi_v(x)$ and $\Psi_v^{-1}=\hat{\phi}_v: P(V)\setminus P(T_v) \rightarrow T_v$ is the wanted coordinate map.

Given non-zero $v$ we want to describe local coordinates of $P(V)$ in a neighborhood of the projective point $[v]={\Bbb C}v$. We do so through the orthogonal complement: $T_v=\{z: \langle a,v \rangle=0\}$.

Except for projective points in $P(T_v)$ we will expres any $[x]$ in the form $[v+w]$ for a suitable $w\in T_v$. To get this map let $x\in V\setminus T_v$ and let us seek $\lambda$ and $w\in T_v$ for which $$ x = \lambda (v+w) $$ Taking scalar product with $v$ we get: $\langle x,v\rangle = \lambda \langle v,v\rangle$ from which we first isolate $\lambda$ and insert in the first to obtain $w$: $$ \phi_v(x):= w = \frac{\langle v,v\rangle}{\langle x,v\rangle}x-v. $$ Now, for any non-zero $\mu$ we have $\phi_v(\mu x)=\phi_v(x)$ so this lifts to a well-defined map: $\hat{\phi}_v([x]) = \phi_v(x)$ and $\Psi_v^{-1}=\hat{\phi}_v: P(V)\setminus P(T_v) \rightarrow T_v$ is the wanted coordinate map.

Exercise: Given $v,v'$ and corresponding $[x]\in A_v\cap A_{v'}$ find the change of coordinate map taking $w\in T_v \mapsto w'\in T_{v'}$

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It is a bit indirect, but there is also a notion of representative onGiven non-zero $T_[v]P(V)$ with respect$v$ we want to describe local coordinates of $P(V)$ in a neighborhood of the baseprojective point $[v]={\Bbb C}v$. If you scaleWe do so through the orthogonal complement: $T_v=\{z: \langle a,v \rangle=0\}$. $v$ by

Except for projective points in $P(T_v)$ we will expres any $[x]$ in the form $[v+w]$ for a suitable $w\in T_v$. To get this map let $x\in V\setminus T_v$ and let us seek $\lambda$ then you also scaleand $a$$w\in T_v$ for which $$ x = \lambda (v+w) $$ Taking scalar product with $v$ we get: $\langle x,v\rangle = \lambda \langle v,v\rangle$ from which we first isolate $\lambda$ and insert in the first to obtain $b$. Note that$w$: $$ \phi_v(x):= w = \frac{\langle v,v\rangle}{\langle x,v\rangle}x-v. $$ Now, for any non-zero $|\lambda|^2$ is real$\mu$ we have $\phi_v(\mu x)=\phi_v(x)$ so it disappears inthis lifts to a well-defined map: $\hat{\phi}_v([x]) = \phi_v(x)$ and $\Psi_v^{-1}=\hat{\phi}_v: P(V)\setminus P(T_v) \rightarrow T_v$ is the ratiowanted coordinate map.

It is a bit indirect, but there is also a notion of representative on $T_[v]P(V)$ with respect to the base point. If you scale $v$ by $\lambda$ then you also scale $a$ and $b$. Note that $|\lambda|^2$ is real so it disappears in the ratio.

Given non-zero $v$ we want to describe local coordinates of $P(V)$ in a neighborhood of the projective point $[v]={\Bbb C}v$. We do so through the orthogonal complement: $T_v=\{z: \langle a,v \rangle=0\}$.

Except for projective points in $P(T_v)$ we will expres any $[x]$ in the form $[v+w]$ for a suitable $w\in T_v$. To get this map let $x\in V\setminus T_v$ and let us seek $\lambda$ and $w\in T_v$ for which $$ x = \lambda (v+w) $$ Taking scalar product with $v$ we get: $\langle x,v\rangle = \lambda \langle v,v\rangle$ from which we first isolate $\lambda$ and insert in the first to obtain $w$: $$ \phi_v(x):= w = \frac{\langle v,v\rangle}{\langle x,v\rangle}x-v. $$ Now, for any non-zero $\mu$ we have $\phi_v(\mu x)=\phi_v(x)$ so this lifts to a well-defined map: $\hat{\phi}_v([x]) = \phi_v(x)$ and $\Psi_v^{-1}=\hat{\phi}_v: P(V)\setminus P(T_v) \rightarrow T_v$ is the wanted coordinate map.

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It is a bit indirect, but there is also a notion of representative on $T_[v]P(V)$ with respect to the base point. If you scale $v$ by $\lambda$ then you also scale $a$ and $b$. Note that $|\lambda|^2$ is real so it disappears in the ratio.