TITLE: Rationality of conic bundles
QUESTION [5 upvotes]: 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 ?

REPLY [6 votes]: 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.