Let $X$ be a normal, $\mathbb{Q}$-factorial complex projective variety, and let $A$, $B$ be two Weil divisors on $X$. From my understanding, the associated sheaves $\mathcal{O}_X(A)$, $\mathcal{O}_X(B)$ are coherent and reflexive.

Define $$P= \operatorname{\mathbf{Proj}} \bigoplus_{m\geq 0} S^m\text{H}^0(X,\mathcal{O}_X(A)\oplus \mathcal{O}_X(B)).$$

That is, I'm mimicking the construction of a $\mathbb{P}^1$-bundle over $X$, but in this case by construction it is not locally trivial.

**Question:** Under which conditions (if any), is $P$ a fibration over $X$ with fibers $\mathbb{P}^1$? Do you happen to know some references for such a construction (or something similar)?

I'm aware I could simply consider $a$, $b$ such that $aA$ and $bB$ are Cartier, and then consider $\mathbb{P}(\mathcal{O}_X(aA)\oplus \mathcal{O}_X(bB))$, but it's not the road I'd like to take.