# Current progress on rationality problem for complex hypersurfaces

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?

• A very recent excellent overview "On rationality problems" by Debarre: perso.imj-prg.fr/olivier-debarre/wp-content/uploads/sites/34/… No smooth rational hypersurfaces of degree 4 and higher are known, and this is one of the key questions in the field, alongside rationality of cubic fourfold. Welcome to MathOverflow! Commented Dec 31, 2022 at 17:15

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.