Assume we have two standard conic bundles $\pi:C \rightarrow X$ and $\pi': C'\rightarrow X'$. That is $\pi$ and $\pi'$ are flat morphims of smooth varieties over $\mathbb{C}$ and both maps are relatively minimal, i.e. the preimage of an irreducible divisor in $X$ resp. $X'$ is an irreducible divisor in $C$ resp. $C'$.
According to [1, 5.3.] these conic bundles correspond to rank 4 sheaves of maximal orders $\mathcal{A}$ and $\mathcal{A}'$ over $X$ resp $X'$ such that the degeneration divisors of the conic bundles coincide with the branch (ramification) divisors of the orders and satify $sing(C)=\bigcup\limits_{i,j} (C_i\cap C_j)$ where $C_i$ and $C_j$ are nonsingular transversally intersecting components of $C$.
Looking at the generic points in $X$ and $X'$ we get two quaternion algebras $A$ and $A'$ over the function fields $\mathbb{C}(X)$ and $\mathbb{C}(X')$, with classes $a$ and $a'$ in the Brauer group of $Br(\mathbb{C}(X))$ resp $Br(\mathbb{C}(X')$.
$\underline{\text{Question}:}$ If we have a birational map $\phi: X \dashrightarrow X'$, that ist $\mathbb{C}(X)\cong \mathbb{C}(X')$, which maps the class $a$ to the class of $a'$, can we somehow construct a birational map $\psi: C\dashrightarrow C'$ such that we have a commutative diagramm $\phi\circ\pi = \pi'\circ\psi$ (that is where theses maps are defined)?
[1] On conic bundle structures - V.G.Sarkisov
