Let $X$ be a smooth cubic threefold over $\mathbb C$, I'm looking for a proof of the following result:
Claim: For a general point $x\in X$, there are exactly 6 lines passing $x$.
It is mentioned in page 328 of Clemens-Griffiths's The Intermediate Jacobian of the Cubic Threefold, but there is no proof (please let me know if I am wrong).
Here is my thought: Since $x$ is general, it's tangent hyperplane $T_xX$ intersects $X$ has a nondegenerated node at $x$ as its only singularity, so $Y=T_xX\cap X$ is a cubic surface with a node.
Moreover, it is known that $Y$ has exactly 6 lines passing through the node (for example, see this post). Therefore these lines should be the 6 lines that we are looking for. However, how can we argue that there are no other lines through $x$ not lying on the tangent hyperplane?
Best Answer
If $L \subset X$ is a line through $x \in X$ then $T_xL \subset T_xX$, hence the line $L$ is tangent to the tangent hyperplane at $x$. But if a line is tangent to a hyperplane, it is contained in it.