Past exam of the short options course of the University of Oxford / 2023 / 3 / a

A projective conic in ${\mathbb{P}}^2(\mathbb{C})$ is the zero locus of a nonzero homogeneous quadratic polynomial, up to nonzero scaling of that polynomial. Under an invertible linear change of homogeneous coordinates, its symmetric matrix has rank $3$, $2$, or $1$. Over $\mathbb{C}$, the corresponding forms are a nonsingular conic such as $XZ-Y^2=0$, two distinct lines such as $XY=0$, or a double line $X^2=0$. In the last case the repeated factor records multiplicity, even though the underlying point set is one line.

A projective line is the projectivization of a two-dimensional linear subspace of ${\mathbb{C}}^3$, equivalently the zero locus of one nonzero linear form. For two such subspaces $U,W$, the dimension formula for a sum of subspaces gives $\dim (U\cap W)\ge 2+2-3=1$. A nonzero vector in the intersection therefore supplies a common projective point. Distinct lines have exactly one such point.