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Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$.

Can we make hypotheses on $X$ and $\Delta$ ensuring that $X$ is rational or unirational ?

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  • $\begingroup$ Just to be sure: by $3$-fold conic bundle, do you mean that the fibres are three-dimensional conics, or that $X$ is a $3$-dimensional space? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 4, 2017 at 2:53
  • $\begingroup$ I mean that $X$ has dimension $3$. The fibers are curves. $\endgroup$
    – Puzzled
    Commented Apr 4, 2017 at 3:06
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    $\begingroup$ Oh yeah, in higher dimension they're usually called quadrics instead of conics. Anyway, this paper by Hassett–Kresch–Tschinkel contains a lot of results in this direction. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 4, 2017 at 3:35
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    $\begingroup$ @gbp. According to a conjecture of Fano (that I have never seen written, but I have seen it mentioned in writings of others), for a conic bundle over $\mathbb{P}^2$ with discriminant of sufficiently high degree and very general, the total space of the conic bundle is not unirational. $\endgroup$
    – Jason Starr
    Commented Apr 4, 2017 at 9:31
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    $\begingroup$ Regarding rationality, as in Clemens-Griffiths you would need the polarized Prym variety of the double cover $\widetilde{\Delta}\to \Delta$ to be a product of polarized Jacobians of curves (as a polarized Abelian variety). $\endgroup$
    – Jason Starr
    Commented Apr 4, 2017 at 9:34

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A fact that has been recently studied is the stable irrationality of $X$, namely the fact that the product $X\times \mathbf{P}^n$ is not rational for any $n$. This, as you might know, implies irrationality as well. You can find out more in the works of Colliot-Thélène and Pirutka; a good survey is, for instance, http://cims.nyu.edu/~pirutka/survey.pdf

The main arguments used involve unramified cohomology and Brauer groups. Basically, one knows that the unramified cohomology groups $H_\mathrm{rm}^2 (k(X)/k,\mathbf{Z}/2)\simeq \mathrm{Br}(X)[2]$ are related to stable rationality. In particular, if $\mathrm{Br}(X)[2]$ does not vanish, then $X$ is not stably rational.

In the above survey paper a formula for the unramified cohomology of conic bundles of the form $\pi : X\longrightarrow \mathbf{P}^2$ is given, attributed to Colliot-Thélène. This formula employs geometric conditions on the discriminant locus to determine the behaviour of $\mathrm{Br}(X)[2]$.

A similar formula, but for conic bundles over threefolds with some "quasi-rational" conditions, was given in the paper https://arxiv.org/abs/1610.04995 by Auel, Boehning, von Bothmer and Pirutka. The formula addresses the unramified cohomology groups of conic bundles of the form $\pi : X\longrightarrow B$, where $B$ is a smooth projective 3-fold with $H^3_{\mathrm{ét}}(B,\mathbf{Z}/2)=0$ and $\mathrm{Br}(B)[2]=0$, and again uses heavily the discriminant locus to describe its behaviour.

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By Corollary 5.6.1 here

https://arxiv.org/pdf/1712.05564.pdf

if $\text{deg}(\Delta)\leq 4$ then $X$ is rational.

If $\text{deg}(\Delta) = 5$ then $X$ could be rational or not depending on whether the double cover $\widetilde{\Delta}\rightarrow\Delta$ is defined by an even or an odd theta characteristic. For instace, by blowing-up a line in a smooth cubic $3$-fold in $\mathbb{P}^4$ we get a conic bundle, with discriminant of degree five, that is unirational but not rational.

By Theorem 9.1 of the same paper if $\text{deg}(\Delta)\geq 6$ then $X$ is not rational. However, by Theorem 7 here

https://arxiv.org/pdf/1712.05564.pdf

or Corollary 1.2 here

https://arxiv.org/pdf/1403.7055.pdf

if $\text{deg}(\Delta)\leq 8$ then $X$ is unirational.

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  • $\begingroup$ Does anyone know why this preprint of Mella is not published, and does not appear in the author's publication list? $\endgroup$
    – abx
    Commented Aug 9, 2021 at 18:08
  • $\begingroup$ Because all the results in Mella's paper (who works over an algebraically closed field of characteristic zero) are proved in much more generality (basically over an arbitrary field with very few exceptions) in his joint paper with Kollar. In this second paper they look at $3$-fold conic bundles over a rational surface as surface conic bundles over the function field of $\mathbb{P}^1$. $\endgroup$
    – Puzzled
    Commented Aug 9, 2021 at 18:55
  • $\begingroup$ Oh, I see, thanks! $\endgroup$
    – abx
    Commented Aug 9, 2021 at 19:05

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