= Hessian curve of a plane curve
{c}
{title2=$H_F=\det\operatorname{Hess}F=0$}
= Hessian curve
{c}
{synonym}
For a <homogeneous polynomial> $F$ of degree $d\geq2$, the determinant of its three-by-three <Hessian matrix> is homogeneous of degree $3(d-2)$ if nonzero. Its zero locus is the Hessian curve; it can be reducible or degenerate, and a nonzero constant for $d=2$ has empty zero locus. Over $\mathbb C$, its intersections with the nonsingular locus of $V(F)$ are precisely the <flexes>, by the <Hessian criterion for a flex>, as long as the tangent is not a component.
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