General linear position

Points in ${\mathbb{P}}^n$ are in general linear position when no $k+1$ of them lie in a projective subspace of dimension less than $k$ for $k\le n$. Equivalently, every selection of at most $n+1$ representing vectors is linearly independent.

Distinct lines in the projective plane are in general position when no three are concurrent. Their union is a reduced plane curve with exactly $\left(\genfrac{}{}{0ex}{}{d}{2}\right)$ singular points, all ordinary pairwise crossings, if there are $d$ lines. An explicit family is $y=t_{i}x+t_i^{2}z$ for distinct complex $t_i$: the intersection of the $i$th and $j$th lines is $[-t_i-t_j:-t_i{t}_j:1]$, and substituting it into a third equation gives $-(t_k-t_i)(t_k-t_j)\ne 0$.