= Cubic surface line incidence variety
{title2=$I=\{(L,[F]):F|_L=0\},\quad\dim I=19$}
A <projective bundle> with fibre $\mathbb P^{15}$ over the four-dimensional <Grassmannian> $\operatorname{Gr}(2,4)$. The condition of containing a fixed line imposes four independent linear conditions on the twenty cubic coefficients. The projection to $\mathbb P^{19}$ is proper. The line $\{x_2=x_3=0\}$ is an isolated point over $x_0^2x_2+x_1^2x_3$; <isolated fibre point forces dominance in equal dimensions> then proves that every cubic surface contains a line.
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