# Line intersecting three lines in $\mathbb{P}^4$ that are not in one hyperplane

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

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

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?

• 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. – The Coding Wombat Mar 28 at 21:19