Smooth quadric surface
= Smooth quadric surface
A <smooth quadric surface> is a smooth two-dimensional <quadric hypersurface>. Over an <algebraically closed field> of <characteristic> other than two, every smooth quadric surface in $\mathbb P^3$ becomes
$$
V(x_0x_3-x_1x_2),
$$
after a <projective linear transformation>. The <Segre embedding> identifies it with $\mathbb P^1\times\mathbb P^1$.