Birational rigidity and K-stability of Fano hypersurfaces with ordinary double points

Abstract

Extending previous results, we prove that for $n\ge 5$ all hypersurfaces of degree $n+1$ in ${\mathbb{P}}^{n+1}$ with isolated ordinary double points are birational superrigid and K-stable, hence admit a weak Kähler--Einstein metric.

v4: 15 pages, last version following referee's report - to appear in the special volume on "Rationality problems" of the Rend. Circ. Mat. Palermo