Current progress on rationality problem for complex hypersurfaces

Question

How is the current progress on rationality problem for complex hypersurfaces $X\subset\mathbb{P}^{n+1}$ with $n\geq 3$?

There are many hypersurfaces are shown to be unrational, such as smooth cubic threefolds and smooth quartic threefolds. Hypersurfaces with degree $d$ large enough are known to be unrational (they are not even uniruled when $d>n+1$).

However, there are only a few examples of rational hypersurfaces.

• A singular cubic hypersurface which is not a cone is ratinoal.
• Some special cubic fourfolds are rational (many people). Moreover, a smooth cubic hypersurface $X\subset\mathbb{P}^{2n+1}$ containing two skew linear spaces of dimension $n$ is rational.

Are there any new rational hypersurfaces in recent years? Or did I miss some known examples?

Answer 1

For upper bounds, there is the paper https://arxiv.org/abs/1801.05397 of Schreieder, which shows (over any uncountable field of characteristic not equal to two) that for $N>2$, a very general $N$-dimensional hypersurface of degree at least $\log_2 N+2$ is irrational (in fact, even stably so.)

On the other hand, for smooth complex hypersurfaces, I am fairly confident that there are still no examples of 1. rational cubic hypersurfaces of odd dimension or 2. rational hypersurfaces with degree $>4$. There are many constructions of rational cubic hypersurfaces of even dimension.

(P.S: I personally do not even know of a odd-degree unirational parametrization of an odd-dimenisonal cubic hypersurface - there are many different natural approaches to construct one, but they all seem to fail for one reason or another...)