I was reading an article, and was reading a proof that I am not quite sure is right. The setup is, that we are considering $k$ points $P_1,\ldots, P_k\in PG(k-1,q)$ in general position, and we then denote $\ell_{i,j}= \langle P_i,P_j\rangle$ for $1\leq i <j \leq k$, and define $M=\bigcup_{i,j}\ell_{i,j}$, and then wish to consider $|M\cap H|$ where $H$ is any fixed hyperplane in $PG(k-1,q)$. This intersection is only dependent on some integer $r$ which is the number of points $P_1,\ldots,P_k$ which are in $H$, and w.l.o.g. we can assume that $P_1,\ldots,P_r\in H$ and $P_{r+1},\ldots P_k\notin H$. We then have
- $M$ and $H$ contains the lines $\ell_{i,j}$ for $1\leq i < j \leq r$
- $H$ intersects $\ell_{i,j}$ in $P_i$ for $1\leq i \leq r <j \leq k$
- $H$ intersects $\ell_{i,j}$ in some pairwise distinct points $Q_{i,j}$ for $r<i<j\leq k$
The article then states based on the above, that
$$ |M\cap H| = \left|\bigcup_{1\leq i \leq r < j \leq k}\ell_{i,j} \right| + \left|\bigcup_{r+1\leq i <j \leq k} Q_{i,j}\right| = \binom{r}{2}(q-1)+r-\binom{k-r}{2} $$
But wouldn't it be necessary to count the number of points on the lines that are contained in both $M$ and $H$, minus the points from the intersections of these lines?