# Degree 2 plane curves on a cubic surface singular along a line in $\mathbb{P}^3$

Suppse $$X$$ is an irreducible cubic surface in $$\mathbb{P}^3$$ singular along a line $$l_1$$. Then clearly there is a plane $$H$$ containing $$l$$ such that $$X \cap H = l_1^2l_2$$. My question is: can $$X$$ contain a degree $$2$$ plane curve other than curves of the form $$l_1l$$, where $$l$$ is some other line ?

• Yes. $X$ is the projection of a cubic scroll $S\subset \mathbb{P}^4$ from a point $p\in \mathbb{P}^4\smallsetminus S$. This $S$ can be viewed as the image of the map $\mathbb{P}^2\rightarrow \mathbb{P}^4$ given by the linear system of conics passing through a fixed point. A general line in $\mathbb{P}^2$ maps to a conic in $\mathbb{P}^4$, which projects isomorphically to a conic in $X$. – abx yesterday