On the decomposition of the diagonal for n-fold hypersurfaces via cubic (n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docu

Abstract

We study the question of the existence of a decomposition of the diagonal for very general degree d⩾4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\geqslant 4$$\end{document} and (2,d⩾3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2,d\,{\geqslant }\, 3)$$\end{document}-complete intersection n-folds. Using cycle-theoretic techniques of Lange, Pavic and Schreieder, we reduce the question to the existence of such a decomposition for (n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-1)$$\end{document}-dimensional cubic hypersurfaces. This extends results of Nicaise and Ottem from stable rationality to the existence of a decomposition of the diagonal. A key ingredient is the essential dimension of cubic hypersurfaces due to Voisin. 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 retract irrationality 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. As another application we show that a very general quintic fourfold containing a three-dimensional linear subspace with multiplicity 2, which is unirational due to results of Massarenti, does not admit a decomposition of the diagonal and is therefore not retract rational.