 Incidence proof that a cubic surface contains a line

Cubic forms in four variables form ${\mathbb{P}}^{19}$. Lines form the four-dimensional Grassmannian $G=\mathrm{Gr}(2,4)$. The incidence pairs $(\ell ,[F])$ with $F{|}_{\ell }=0$ form a projective-space bundle with fibre ${\mathbb{P}}^{15}$ over $G$, since restriction to a line imposes four independent linear conditions. Hence the incidence variety is irreducible of dimension 19. For $F=x_0^2{x}_2+x_1^2{x}_3$, a nearby line written $x_2=a_0{x}_0+a_1{x}_1$, $x_3=b_0{x}_0+b_1{x}_1$ lies on $F$ precisely when all four parameters vanish. This is an isolated point of that fibre. The fiber dimension theorem forces the image to have dimension 19, and projectivity makes the image closed. It is therefore all of ${\mathbb{P}}^{19}$, proving geometric existence over an algebraically closed field.