Let $X\subset \mathbb P^4$ be a smooth cubic threefold, $\{S_t\}_{t\in \mathbb P^1}$ be a generic pencil of hyperplane sections (so each one is a cubic surface, for smooth one it contains $27$ lines).
I read from somewhere the following statement:
This defines a degree $27$ cover $\pi: C\to \mathbb P^1$ with $24$ branch points in $\mathbb P^1$, and over each of the $24$ points, the fiber consists of $6$ points of multiplicity two and $15$ single points.
I can understand the number of branch points $24$ is because the degree of dual variety $\check{X}$ is $24$. But why such fiber consists $6$ points of multiplicity two?
Everything is over $\mathbb C$.