Conics in Complex Projective Spaces

Question

I was reading classification of complex hyperquadrics, I am stuck in $\mathbb CP^2$ what is $X_0^2+X_1^2=0$ in $\mathbb CP^2$, ok in $\mathbb CP^1$ this represents just two points, my attempt if $X_1$ or $X_0$ is zero then we get a point [$0:0:1$], if non zero we get [$i:1:X_2$] and [$-i:1:X_2$] is it also $\mathbb CP^1$? and if $X_2=0$ what do we get?

Answer 1

Note that over the complex numbers you have a decomposition $$ X_0^2+X_1^2=(X_0+iX_1)(X_0-iX_1). $$ Thus your "conic" actually breaks up in the union of the two algebraic subsets $$ X_0\pm iX_1=0 $$ which are lines (each isomorphic to ${\Bbb P}^1(\Bbb C)$). Thus the name "degenerate conic" which one usually associates with this situation.