Rationality of conic bundles 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 ?
 A: 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.
A: 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.
