Projectivization of direct sum of line bundles

Question

Is there a way to understand the projectivization of direct sum ( finite if needed ) of line bundles in terms of $\mathbb{P}^1$- bundles under appropriate conditions?

Thanks in advance!

Answer 1

Let me explain the rank 3 case. Let $L_1$, $L_2$, and $L_3$ be a triple of line bundles on a scheme $S$. Then there is a natural rational map $$ \mathbb{P}_S(L_1 \oplus L_2 \oplus L_3) \dashrightarrow \mathbb{P}_S(L_1 \oplus L_2) \times_S \mathbb{P}_S(L_2 \oplus L_3). $$ To make it regular one should blowup two sections $\mathbb{P}_S(L_1)$ and $\mathbb{P}_S(L_3)$ on $\mathbb{P}_S(L_1 \oplus L_2 \oplus L_3)$ and then contract the strict transform of the relative hyperplane $\mathbb{P}_S(L_1 \oplus L_3)$. Note that this is a relative version of a standard toric transformation between $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$.

In higher rank the situation is similar --- there is a rational map to the fiber product of $\mathbb{P}^1$-bundles, which can be resolved by means of toric geometry.