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Assume $X$ is a smooth projective variety over $\mathbb{C}$ of dimension $n$, here $n\geq 3$, with a reduced normal crossing divisor $D\subset X$, such that $D=\sum\limits_{i=1}^r D_i$ where the $D_i$ are the irreducible and nonsingular components of $D$ and $sing(D)=\bigcup\limits_{i\neq j}(D_i\cap D_j)$.

  • Can we find a regular conic bundle $p: Y\rightarrow X$ with discriminant divisor $\Delta=D$?

That is: $Y$ is a smooth projective variety of dimension $n+1$, $p$ is a flat morphism with the following type of fibres: for $x\in X\backslash D$ we have $p^{-1}(x)$ is a smooth conic $C$. For $x\in D\backslash sing(D)$ we get $p^{-1}(x)=\mathbb{P}^1$v$\mathbb{P}^1$, two lines meeting in one point. And finaly for $x\in sing(D)$ we have $p^{-1}(x)=2\mathbb{P}^1$ a nonreduced double line.

If possible, how would one construct such a conic bundle? I guess one should try to find a rank 3 locally free sheaf $\mathcal{E}$ on $X$ and the try to find an irreducible divisor $Z\subset \mathbb{P}(\mathcal{E})$ with the desired properties. But how can one do that?

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No. There is a reciprocity law at play here which places additional restrictions on the discriminant locus.

The precise relations are a bit complicated and are easier to phrase in terms of the corresponding Brauer group element. See e.g. Theorem 1 from Section 3 of "Artin, Mumford - Some elementary examples of unirational varieties which are not rational" for the case of conic bundles over a surface.

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