Let $f \in k[x_0, \dots, x_3]$ be a homogeneous cubic polynomial that defines a smooth hypersurface in $\mathbb{P}^3$ and let $H(f) = \det \left[ \frac{\partial^2 f}{\partial x_i \partial x_j}\right]_{ij}$ be the degree $4 \cdot (3-2)=4$ hessian hypersurface of $f$. Is the number of intersections between $f$ and $H(f)$ non-zero?
If $H(f)$ is smooth and $H(f)$ and $f$ intersect transversely, then by Bezout's theorem we would get 12 intersection points, but I don't see why these conditions should hold.
The context of this question is Exercise 7.3(iv) of Undergraduate Algebraic Geometry by Reid, where we prove that a smooth cubic surface has a cuspidal cubic section.
In this answer Can two projective surfaces intersect in points only? a reference to Hartshorne is given, where it is proven that if two varieties have dimension $r,s$, then their intersection has dimension $\geq r+s -n$, and in addition if $r+s-n \geq 0$, then their intersection is non-empty. Thus in this case $r = s = 2$ so the intersection dimension is at least $4-3 = 1$. So the intersection is non-empty.