Smooth quadric surface

Over an algebraically closed field of characteristic other than two, every smooth quadric surface in ${\mathbb{P}}^3$ becomesafter a projective linear transformation. The Segre embedding identifies it with ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

For $X=\{x_0{x}_1=x_2{x}_3\}$ and $P=[1:0:0:1]$, the displayed map is defined on $X\setminus \{P\}$. Its fibre over $[u:v:w]$ has equation $(u-v)\lambda +uw=0$ on an affine line. Away from $u=v$ the fibre is one reduced point. On that line it is empty except at $[0:0:1]$ and $[1:1:0]$, where it is a punctured ruling line, hence an affine line. The two exceptional lines are the rulings of a smooth quadric surface through $P$.

Over an algebraically closed field, remove a smooth hyperplane conic $C$ from a smooth quadric surface $Q$. The rulings of a smooth quadric surface generate $\mathrm{Pic}(Q)={\mathbb{Z}}^2$, and $C$ has class $(1,1)$. Picard-group localization on a smooth variety gives the quotient by $\mathbb{Z}(1,1)$. The restricted ruling bundle ${\mathcal{O}}_Q(1,0)$ is a generator, exhibiting a nontrivial line bundle on an affine surface.

Over an algebraically closed field, a smooth four-variable projective quadratic equation can be written $x_0{x}_3-x_1{x}_2=0$, including in characteristic two. Its quadratic form splits into two hyperbolic planes: choose an isotropic vector and its paired vector, adjust the latter to be isotropic, and repeat on the orthogonal complement. In characteristic two, smoothness forces the alternating polar form in dimension four to have full rank, since a positive-dimensional radical has a nonzero isotropic vector over the algebraically closed field and would give a singular point. The Segre embedding $([s_0:s_1],[t_0:t_1])\mapsto [s_0{t}_0:s_0{t}_1:s_1{t}_0:s_1{t}_1]$ then gives the isomorphism. Each matrix of coordinates has rank one, which supplies the inverse pair of projective factors.

Under $Q\cong {\mathbb{P}}^1\times {\mathbb{P}}^1$, the fibres of the two coordinate projections are the two rulings of the smooth quadric surface. Two distinct fibres in one ruling are disjoint projective lines, while one fibre from each ruling meets in one point.

For distinct $p,q\in {\mathbb{P}}^1$, the curves ${\mathbb{P}}^1\times \{p\}$ and ${\mathbb{P}}^1\times \{q\}$ are disjoint, smooth, and projective.