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It is a continue of the previous question twisted cubic in a hyperplane section of a cubic threefold. Let $X$ be a complex smooth cubic threefold and $C$ be a smooth twisted cubic, then $C\subset Y\subset X$ for a unique cubic surface $Y$ in $X$.

When $Y$ is smooth, we have 27 lines of second type on $Y$: one has $$\operatorname{Ext}^1(\mathcal{O}_L(-1),\mathcal{O}_L(1))=0$$ and $$0\rightarrow\mathcal{N}_{L/Y}=\mathcal{O}_L(-1)\rightarrow\mathcal{N}_{L/X}\rightarrow\mathcal{N}_{X/Y}|_L=\mathcal{O}_L(1)\rightarrow0$$ The twisted cubic $C$ misses 6 of them, intersects 15 of them once and 6 of them twice.

What about the lines on a singular hyperplane section $Y$?

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Such a hyperplane section is a normal integral cubic surface. Such cubic surfaces are classified by Bruce and C. T. C. Wall in On the Classification of Cubic Surfaces.

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  • $\begingroup$ If this satisfactorily answers your question, then don't forget to accept it. $\endgroup$
    – LSpice
    Commented May 6, 2023 at 12:26

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