Let $\mathcal{L}$ be a pencil of cubics on $\Bbb{P}_\Bbb{C}^2$ whose general member is smooth and such that all members are irreducible.
I'm trying to prove/disprove the following:
The base locus of $\mathcal{L}$ consists of nine $9$ distinct points.
I do have an intuitive argument, but formally speaking I'm still lost.
The argument is: let $C,C'$ be smooth cubics. Generally, $C,C'$ meet in $9$ distinct points $P_1,...,P_9$ in the most general position possible (i.e., no three points are in a line, no six are in a conic). Let $\mathcal{L}'$ is the pencil of cubics through $P_1,...,P_9$. The fact that $\mathcal{L}$ has only irreducible curves should mean that it is general in some sense, therefore essentially like $\mathcal{L}'$.
I've tried to formalize this by looking at $\Bbb{P}^9$ as the space of all cubics in $\Bbb{P}^2$ and $\mathcal{L}$ as a general line in it. But I don't know how to relate this with the condition of $9$ distinct points.
Any help is appreciated, thank you!