The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved.
In this talk I will discuss rationality of forms of these Fanos over nonclosed fields of characteristic 0.
I will concentrate on the case where the Picard number equals 1.
The talk is based on joint work with Alexander Kuznetsov (in progress).
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/5312