I want to compute the homology of $M=Z(x_0^2+x_1^2+x_2^2)\subset \mathbb{C}P^2$. I think I have the answer, but I'm not sure how to make it precise.
My approach is to consider the affine cover $U_0=Z(x_0)^c$. Then under the homeomorphism $\varphi_0:[x_0:x_1:x_2]\mapsto (\frac{x_1}{x_0},\frac{x_2}{x_0})$ we find $$\varphi_0(M)=\{(x_1,x_2)\in\mathbb{C}^2\mid 1+x_1^2+x_2^2=0\}$$ Which we recognize as a Riemann surface. Now we know that we can compactify this Riemann surface by adding $2$ points at infinity, and the result will have genus 0. I suspect that these are exactly the points in $$M\setminus U_0=\{[0:x_1:ix_2],[0:x_1:-ix_2]\}$$ And so we find that $M$ is homeomorphic to a sphere, which tells us its homology.
My problem is that I do not see how to incorporate these 2 missing points into the picture in a more precise\rigorous way.
Why not directly prove it is isomorphic to the projective line? For example, changing variables to $(x_0+ix_1, x_0-ix_1,ix_2)$, this curve $M$ is same as the one given by $x_0x_1-x_2^2=0$ and you have a map $\mathbb{P}^1\to M$, by $(u,v)\mapsto (u^2, v^2, uv)$, which you can easily see to be an isomorphism.