I know that every rational elliptic surface is given by the blowup of the $9$ points of intersection of two cubics in $\Bbb{P}^2$ (at least one of them being smooth).
Take for example the cubics $C_1,C_2\subset\Bbb{P^2}$ such that $C_1$ is given by $yz^2-x(x^2-z^2)=0$ and $C_2$ is the triple line $3L$ where $L:y=0$, and let $\pi:X\to \Bbb{P}^1$ be the elliptic surface generated by them.
(this is just a particular example I'm curious about, but any other will be fine)
What I'm trying to do is to identify all singular fibres of $\pi$ and classify them by their Kodaira symbols.
I know how to do that given the Weierstrass equation for the surface, but I don't know how to find it here.
I've stumbled upon this nice material which gave me the impression that these things can be treated geometrically, with actual pictures, but I don't understand how the constructions are made.
In the particular example I gave, I don't know what happens to the triple line after the blowup and what information this will give about the bad fibers. Even in simpler cases, like when $C_1, C_2$ meet in $9$ different points, I do see what happens in after blowing up, but I still don't understand what that tells me about the bad fibres.
I wish I could understand this geometric approach, since this way it would be interesting to come up with other different possibilities for $C_1,C_2$ and see what happens.
Any suggestions?