= Hessian curve of a plane cubic
{c}
{title2=$\det\operatorname{Hess}F=0$}
For a cubic <homogeneous polynomial> $F$, its <Hessian matrix> has linear entries, so its determinant is a cubic <homogeneous polynomial>, possibly zero. On a <nonsingular plane cubic> in characteristic different from two, its zeros are exactly the <flexes>, by the <Hessian criterion for a flex>. If the determinant is nonzero, <Bézout's theorem> supplies intersections; if it is zero, the local criterion itself supplies flexes.
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