On the diagonal of quartic hypersurfaces and $(2,3)$-complete intersection $n$-folds

We study the question of the existence of a decomposition of the diagonal for very general quartic and $(2,3)$-complete intersection $n$-folds. Using cycle-theoretic techniques of Lange, Pavic and Schreieder we reduce the question via a degeneration argument to the existence of such a decomposition for $n-1$-dimensional cubic hypersurfaces and their essential dimension. A result of Voisin on the essential dimension of complex cubic hypersurfaces of odd dimension (and of dimension four) then yields conditional statements that extend results of Nicaise and Ottem from stable rationality to the existence of a decomposition of the diagonal. As an application, we use a recent result of Engel, de Gaay Fortman and Schreieder on the decomposition of the diagonal for cubic threefolds to give a new proof of the non-retract rationality of a very general complex quartic $4$-fold, originally due to Totaro, and of a very general complex $(2,3)$-complete intersection $4$-fold, originally due to Skauli.