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Given three lines $L,M,N$ in $\mathbb{P}^4$, not all in one hyperplane, I want to show by example that it is possible that there are multiples lines intersecting $L,M$ and $N$.

What I know: A projective line through two points $P=(p_0:...p_n), Q=(q_0:...q_n)$ is defined by first moving these points to $\mathbb{R}^{n+1}$, and then we have

$$ PQ=<P,Q>=\{\lambda p_0 + \mu q_0 : ... : \lambda x_n + \mu q_n \mid (\lambda,\mu)\neq (0,0)\}. $$

So a line PQ in projective space is a plane through the lines OP and OQ in Euclidian space. (I think)

How do I have to think about these lines (planes?) in $\mathbb{P}^4$ and how would I find lines intersecting all these lines?

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  • $\begingroup$ Would this work? Assume L and M intersect in a point P. By the dimension formula we have that $\langle L, M \rangle \cap N=\emptyset$. Choose any point $Q$ on $N$ and draw a line from $P$ to $Q$. You now have multiple lines. $\endgroup$ – The Coding Wombat Mar 28 at 21:19

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