Hypersurface singularity

A point on $F=0$ where all first partial derivatives vanish. For a reduced hypersurface this is the Jacobian criterion; a nondegenerate quadratic leading part defines an ordinary double point.

Restrict a cubic equation to the affine line through two distinct singular points. At each point the equation and its first directional derivative vanish, so the restricted polynomial has two distinct double roots. A polynomial of degree at most three with this property is identically zero. Hence the whole line lies in the hypersurface. The multiplicity argument works in every characteristic.