Good time of day. I have the following question
Milnor hypersurface $H_{ij}$ is a smooth hypersufrace in $\mathbb CP^i \times \mathbb CP^j$ for fix pair of integers $j \ge i\ge 0$. Its algebraic subvariety
$$H_{ij}= \left\{{z_0: \ldots :z_i)} \times {(\omega_0: \ldots :\omega_j) \in \mathbb CP^i \times \mathbb CP^j: z_0\omega_0+ \ldots +z_i\omega_i=0}\right\} $$
1)I don't know how to prove that $H_{11}$ is homeomorphic to $\mathbb CP^1$
2)And the second question how to prove that $H_{1j}$ is not homeomorphic to $\mathbb CP^1 \times \mathbb CP^{j-1}$? I try to use the following approach based on cohomology rings. I know that cohomology ring of $H^{*}(\mathbb CP^i \times \mathbb CP^j)=Z[x,y]/(x^{i+1}=0, y^{j+1}=0)$. I don't know how to find cohomology ring of milnor hypersurface $H_{1j}$.
If you don't mind ,please, can you explain this in more details? Thank you