Let $\mathbb CP^{n+1}$ be the complex projective space where n is even.
Let S = the subset where $x_0x_1 + x_2x_3 + ... x_{n}x_{n+1} = 0$
Show that S is a (toplogical) manifold and what is its dimension?
I tried to cover by the usual $U_i$ sets where the $i^{th}$ coordinate is non-zero. But I can't show it is homeomorphic to some $\mathbb C^n$. Some help?