Rulings of a smooth quadric surface

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.