Im struggling to wrap my head around an example. It considers the conic $$ x^{2}+y^{2}-z^{2}=0 $$ then proceeds:
Take $A=[1,0,1]$ and the line $P(U)$ defined by $x=0$. Note that this conic and the point and line are defined over any field since the coefficients are 0 or $1 .$ A point $X \in P(U)$ is of the form $X=[0,1, t]$ or $[0,0,1]$ and the map $\alpha$ is
\begin{align} \alpha([0,1, t]) &=[B((0,1, t),(0,1, t))(1,0,1)-2 B((1,0,1),(0,1, t))(0,1, t)] \\ &=\left[1-t^{2}, 2 t, 1+t^{2}\right] \\ \text { or } \alpha([0,0,1])=[-1,0,1] . \end{align}
How do I evaluate $B(v,v)$ or $B(v,v)(a,b,c)$ like they have to go from the first line to the second?
and also why is a point $X \in P(U)$ of the form $X=[0,1, t]$ or $[0,0,1]$
here is the source for reference:https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Algebraic_curves/algebraiccurves2009.pdf
any help would be fantastic!
For the first question, it is exactly evaluation of a bilinear form, given by the matrix B, on the given vectors: so $B(v,w)=v^tBw$. This gives you a scalar which is understood multipling the following vector.
For the second one, when you have a projective line $l$ and two dinstinct points $P,Q$ on it then every other point can be written as $\lambda P+\mu Q$, for $[\lambda,\mu]\in\mathbb{P}^1$. So you can define $t=\mu/\lambda$, if $\lambda\neq 0$, and consider $l$ as the points in the form $ P+tQ$ union the remaining point corresponding to $\lambda = 0$.