We know that the intersection of a smooth cubic surface in $\mathbb{P}^3$ with a plane is a cubic, a line+a conic or three disjoint lines. Can this intersection consist of a conic and its tangent line?



Intersection of a smooth cubic surface with a plane
I guess so. Try the cubic surface whose equation is $$x(xy-z^2)-y^3+w^3=0.$$


If my computations are correct, it is smooth and the intersection with any of the three planes $y- \eta w=0$ (where $\eta$ is a third root of unity) is isomorphic to $x(xy-z^2)=0$, which is the union of a smooth conic and one of its tangent lines.


