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A set of lines is said to be in general position if no $3$ of them meet in the same point. What is the reason of this definition? In $\mathbb{P}^2$, i can understand that two generic lines meet in a single point. But, for example, in $\mathbb{P}^n$ with $n \geq 3$, aren't two generic lines, intuitevely, supposed to not intersect at all? (I imagine two generic lines as being skew). But, according to this definition, two intersecting lines in $\mathbb{P}^n$ with $n \geq 2$ would still be in general position.

Is there a better way to understand the notion of general position, maybe using Zariski topology?

un umile appassionato
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  • Two lines in $\mathbb{P}^n$ always intersect, since their intersection corresponds to the intersection of two planes in $\mathbb{C}^{n+1}$. – Joppy Apr 27 '17 at 10:17
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    Well, two planes through the origin in $\mathbb{C}^{n+1}$, $n \geq 3$, surely may intersect only in the origin. – un umile appassionato May 08 '17 at 15:46

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