Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 24F / b / Solution

Over an algebraically closed field of characteristic different from two, a projective change of coordinates puts every smooth quadric surface in the formThe Segre embedding identifies this surface with ${\mathbb{P}}^1\times {\mathbb{P}}^1$ byTwo distinct fibres of either coordinate projection are disjoint lines in one of the rulings of a smooth quadric surface. Explicitly,both lie on $V$ and have no common projective point. Thus $V$ contains a pair of disjoint lines.