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Let $S$ be a smooth projective variety over $\mathbb{C}$. A conic bundle over $S$ is a smooth projective variety $X$ together with a flat morphism $\pi:X \to S$ all of whose fibres are isomorphic to plane conics.

Let $Y$ be a smooth projective variety equipped with a morphism $\psi:Y \to S$ whose generic fibre is isomorphic to a smooth conic. Does $Y$ admit a conic bundle model?

More specifically, does there exist a conic bundle $\pi:X \to S$ and a birational map $\varphi: X \dashrightarrow Y$ such that $\pi = \psi \circ \varphi$?

Remark: I'm happy with a weakening where there is a regular model where each fibre is a curve, not necessary a conic (i.e. singular fibres may have many irreducible components). But I definitely want my model to be flat over $S$!

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  • $\begingroup$ No, there does not always exist such a conic bundle structure. If you let yourself blow up $S$, then there will always exist a conic bundle structure. If you start with a conic bundle $X'\to S'$ and then consider a contraction $S'\to S$, then for some strict resolution of singularities $Y\to X'$, the composition from $Y$ to $S$ will satisfy your hypotheses. But typically there is no conic bundle structure over $S$ that pulls back over $S'$ to be $S'$-birational to $X'$. The most well-studied example is the "quasi-map contraction" $S$ of the moduli space of stable maps $S'$. $\endgroup$
    – Jason Starr
    Commented Jun 1, 2022 at 10:27
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    $\begingroup$ Thanks Jason! Do you a reference/proof for why a conic bundle structure exists after possibly blowing-up S? $\endgroup$
    – Daniel Loughran
    Commented Jun 1, 2022 at 10:44
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    $\begingroup$ I've found reference: Theorem 1.13 in "V. G. Sarkisov - On conic bundle structures". Thanks! $\endgroup$
    – Daniel Loughran
    Commented Jun 1, 2022 at 11:37

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