I have been looking at sequential blowups of six points $P_i$ in $\mathbb{P}^2$. In the general case, we can identify the blowup with a nonsingular cubic surface in $\mathbb{P}^3$ and under this identification the 27 lines on this surface are
- 6 $E_i$, the exceptional line of $P_i$.
- 15 $F_{ij}$, the strict transform of the line through $P_i$ and $P_j$.
- 6 $G_i$, the strict transform of the conic passing through all $P_j$ except $P_i$.
The thing I have been interested in, is what happens to the $E_i, F_{ij}$ and $G_i$ when the points $P_i$ are in a nongeneral position, for instance is three of them are collinear. And for lines being contracted to one point, I have counted the number of lines passing through this point.
I have done this when all the points are in $\mathbb{P}^2$. But I do not know how to think about it when say $P_2$ is a point on $E_1$. Does it even make sense to talk about the line through $P_1$ and $P_2$?