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 $C⊂Y⊂X$ for a unique cubic surface $Y$ in $X$.

When $Y$ is smooth, we have 27 lines of second type on $Y$: one has $$\text{Ext}^1(\mathcal{O}_L(-1),\mathcal{O}_L(1))=0$$ 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 $Y$?

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 $Y$?

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 $C⊂Y⊂X$ for a unique cubic surface $Y$ in $X$.

When $Y$ is smooth, we have 27 lines of second type on $Y$: one has $\text{Ext}^1(\mathcal{O}_L(-1),\mathcal{O}_L(1))=0$ 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 $Y$?

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 $C⊂Y⊂X$ for a unique cubic surface $Y$ in $X$.

When $Y$ is smooth, we have 27 lines of second type on $Y$: one has $$\text{Ext}^1(\mathcal{O}_L(-1),\mathcal{O}_L(1))=0$$ 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 $Y$?