Two dimensional scrolls contained in quadric cones in P^5

The 2-normality of smooth complex projective varieties is a classical problem in algebraic geometry. In the context of smooth linearly normal surfaces in ${\Bbb P}^5$, an interesting topic is to know which of these surfaces are contained in a quadric hypersurface $\Gamma$. If $\Gamma$ is smooth, it can be identified with the Grassmannian of lines in ${\Bbb P}^3$, and this is a very appropiate tool. If $\Gamma$ is singular, other methods need to be used. Here, the authors prove that a smooth, complex, ruled surface is embedded in ${\Bbb P}^5$ as a linearly normal scroll and contained in a quadric cone of rank five if and only if it is either a rational degree four scroll or an indecomposable elliptic scroll of invariant $e=0$. They also obtain some results on scrolls contained in cones of lower ranks.