How to show $x_0^2+x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \iff \mathbb{CP}^1$

Question

I am currently trying to blow-up an $A_n$ singularity defined by the hypersurface equation: \begin{equation} z_1^2+z_2^2+z_3^{n+1}=0 \subset \mathbb{C}^3 \end{equation}

Let $x_i, i=0,1,2$ denote the homogeneous coordinates on $\mathbb{CP}^2$, then going through the blow-up procedure one gets the hypersurface \begin{equation} x_0^2+x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \end{equation} I know that this is equivalent to $\mathbb{CP}^1$, but I don't know how to show this explicitly.

Similarly, one also gets surfaces of the form for example \begin{equation} x_1^2+x_2^2=0 \subset \mathbb{CP}^2 \end{equation} which should correspond to two $ \mathbb{CP}^1$ intersecting at a point, which again I don't know how to show explicitly.

Any help would be greatly appreciated.

Answer 1

Your first equation $$\tag{1}x_0^2 + x_1^2 + x_2^2 = 0$$ defines a smooth quadratic curve $C$ in $\mathbb{CP}^2$, which is isomorphic to $\mathbb{CP}^1$. To see this, you can verify that the Veronese morphism $$\mathbb{CP}^1 \to \mathbb{CP}^2, [s:t] \mapsto [s^2:st:t^2]$$ gives an isomorphism from $\mathbb{CP}^1$ to $C$.

For $$\tag{2}x_1^2 + x_2^2 = 0$$ note that you have a factorization $x_1^2 + x_2^2 = (x_1 + i x_2)(x_1 - i x_2)$. Each factor defines a line in $\mathbb{CP}^2$, both lines meet in the point $[0:0:1]\in \mathbb{CP}^2$