Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X)=\pi^{*}Pic(S)\oplus \mathbb{Z}K_X$.
In [1, Proposition 2.4] it is proved, that if $S$ is a smooth projective surface and $\sigma: S_1\rightarrow S$ is the blow up at a point $p\in S$, then $\pi_1:X_1\rightarrow S_1$ is a standard conic bundle, here we have $X_1:=X\times_S S_1$. This is done by an analysis of the position of $p$ relative to the discriminant curve $D$, that is one considers the cases $p\notin D$, $p\in D\backslash sing(D)$ and $p\in sing(D)$. In the first case the discriminat on $S_1$ is $\sigma^{-1}(D)$ but in the other two cases it is $\sigma^{-1}(D)\backslash\sigma^{-1}(p)$.
$\underline{Question}$: What happens in higher dimension, i.e. $dim(S)>2$? Can we describe the (or some) smooth irreducible centers $Z\subset S$ such that the blow up $\sigma_Z: S_Z \rightarrow S$ of $Z$ induces a standard conic bundle $\pi_Z: X_Z\rightarrow S_Z$, for $X_Z=X\times_S S_Z$. How does the discriminant curve look like in $S_Z$?
In [2, Lemma 2.2] it is stated, that if $\pi: X\rightarrow S$ is a standard conic bundle with discriminant divisor $D$ and $Z\cap D=\emptyset$, then $\pi_Z: X_Z\rightarrow S_Z$ is a standard conic bundle, with discriminant divisor being $\sigma_Z^{-1}(D)$. There is also a statement about the blow up at a center $Z\subset D$, having the property that the restriction of the double cover of $D$ defined by $\pi$ to $Z$ splits. But this only gives, as far as I understand, a standard conic birationally equivalent to $X_Z$ but does not have to be actually $X_Z$.
Maybe already the case $dim(S)=3$ is of interest. Here the possibilities for $Z$ are a point or a smooth irreducible curve. How do they need to lie relatively to $D$ such that the blow up of $Z$ produces a standard conic bundle? I am not an expert when it comes to blow ups in higher dimension, but I think the main problem will be ensuring that $X_Z$ is smooth. In [1] this is done by some nice geometric arguments, can this be done in higher dimension also?
[1] Sarkisov - Birational automorphisms of conic bundles
[2] Sarkisov - On conic bundle structures
