# Map from a curve to the projective plane

I am working on a exercise and I would like to ask you for your help because I am terrible lost. Let us suppose that $$C\subset \mathbb{P}^2_k:=\text{Proj}(k[x_0,x_1,x_2]$$ is an algebraic curve given by the equation $$x_0^2+x_1^2-x_2^2=0$$, where $$k$$ is an algebraically closed field of characteristic zero. We have a natural map $$\phi$$ from $$C$$ to the set of lines of $$\mathbb{P}^2$$, being this latter space identified with $$\mathbb{P}(H^0(\mathbb{P}^2,\mathcal{O}(1))\simeq \text{Proj}(k[y_0,y_1,y_2])$$, by sending each point $$p$$ of $$C$$ to the line passing through $$p$$ and the infinity point $$(1,0,0)$$. My guess is that working in coordinates, the above map is defined as $$\phi(a,b,c):=(a+1:b:c)$$ But, I would like to know, what is the induced map between the rings, that is $$k[y_0,y_1,y_2]\rightarrow k[x_0,x_1,x_2]/(x_0^2+x_1^2-x_2^2)$$ I would like to know what is the equation of the image $$Y$$ of the map $$\phi$$, and if the morphism $$C\rightarrow Y$$ is a morphism of algebraic curves, and if the morphism induced between its field of fractions $$\Sigma_Y\hookrightarrow\Sigma_X$$ has degree 2. Moreover, is it true that $$\phi^{\ast}\mathcal{O}(1)\simeq\mathcal{O}(2)?$$

Let $$(\mathbb{P}^2)^*$$ denote the projective space of lines in $$\mathbb{P}^2$$. Let $$[y_0:y_1:y_2]$$ be the coordinates of $$(\mathbb{P}^2)^*$$, where $$[y_0:y_1:y_2]$$ denotes the line $$y_0x_0+y_1x_1+y_2x_2=0.$$
If $$p=[x_0:x_1:x_2]\in C$$ then the line spanned by $$p$$ and $$[1:0:0]$$ is $$0*x_0+(x_2)*x_1+(-x_1)*x_2=0,$$ i.e., the map $$\varphi$$ you define sends $$p$$ to $$[0:-x_2:x_1]$$. (Note that $$x_1=x_2=0$$ is impossible since otherwise $$p=[1:0:0]$$ which is not on the curve). The image of this map is the variety $$Z = \{l\in (\mathbb{P}^2)^*\mid [1:0:0]\in l\},$$ i.e., the variety of all lines that passes through a fixed point (which is itself a line in $$(\mathbb{P}^2)^*$$). You may see it in the affine patch $$\mathbb{C}^2\subset\mathbb{P}^2$$ of points with $$x_0\neq 0$$. Then $$C$$ is a hyperbola, $$[1:0:0]$$ is the origin and the corresponding affine patch of the variety $$Z$$ is the set of all lines in $$\mathbb{C}^2$$ that passes through the origin.
The induced map between the rings is $$k[y_0,y_1,y_2]\rightarrow k[x_0,x_1,x_2]/(x_0^2+x_1^2-x_2^2)$$ which maps $$y_0\mapsto 0, y_1\mapsto x_2, y_2\mapsto -x_1$$.