= Hessian criterion for flexes of a plane cubic
{c}
{title2=$F(P)=\det(\partial_i\partial_jF)(P)=0$}
For a smooth homogeneous cubic in characteristic zero, its <plane cubic flexes> are precisely its intersections with the determinant of its <Hessian matrix>. Choose coordinates with a given smooth point $(0:0:1)$ and its tangent $y=0$. Up to a nonzero coefficient, the cubic starts as $yz^2+a x^2z+bxyz+cy^2z$ plus cubic terms in $x,y$. Its tangent contact has order three exactly when $a=0$, and the Hessian determinant at the point is a nonzero constant times $a$. This proves the criterion.
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