Map from a curve to the projective plane

Question

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 $\phi^{\ast}\mathcal{O}(1)\simeq\mathcal{O}(2)?$

Answer 1

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 $y_0\mapsto 0, y_1\mapsto x_2, y_2\mapsto -x_1$.