Given three lines, $L, M, N \in\mathbb{P}^4$, not in one hyperplane and not pairwise intersecting, I need to calculate

$$\dim(\langle L,M\rangle\cap N)$$.

I can however not find a definition for pairwise non-intersecting lines. How do I think about pairwise non-intersecting lines in $\mathbb{P}^4$?


It's simple. $L,M,N$ are pairwise non-intersecting if each pair doesn't intersect; that is, $L\cap M$, $M\cap N$, and $N\cap L$ are all empty.

This principle applies to many other "pairwise X" phrases as well.

  • $\begingroup$ So no line intersects any other line? What does the pairwise add? Why not just say they're not intersecting? $\endgroup$ – The Coding Wombat Mar 27 at 22:08
  • 1
    $\begingroup$ If we said they don't intersect, we would be saying that the triple intersection $L\cap M\cap N$ is empty - a much weaker condition. $\endgroup$ – jmerry Mar 27 at 22:09

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