I am doing some calculation on a toy example from the question here.
Let $\mathbb P(E) \rightarrow \mathbb P^1$ be the projectization of the vector bundle $E = \mathcal O \oplus \mathcal O \oplus \mathcal O(-1) \oplus \mathcal O(1)$ over $\mathbb P^1$.
From direct calculation on the complete linear system $|\mathcal O_{\mathbb P(E) }(4)|$, I found
- The base locus ${\rm Bl} |\mathcal O_{\mathbb P(E) }(4)|$ is the curve $C_0$, where $C_0$ is the image of the composition of $O(-1) \hookrightarrow E \rightarrow P(E)$.
- For a generic divisor $X$ in $|\mathcal O_{\mathbb P(E) }(4)|$ and generic fiber $F$ of $\mathbb P(E) \rightarrow \mathbb P^1$, $X \cap F$ is a surface (in fact, $K3$) with a Du Val singularity at $F \cap C_0$.
Are there some general viewpoints, from which one can predict these phenomenons (the base locus and the type of singularity) without doing actual calculation? I would like to study $|\mathcal O_{\mathbb P(E) }(r)|$ about general $E$ (with $\deg(E)=0$) of higher rank $r$ in more systematic way.
