On the rationality problem for low degree hypersurfaces

TL;DR

The paper develops a cycle-theoretic obstruction to decompositions of the diagonal for low-degree hypersurfaces by extending degeneration techniques to non-strictly semi-stable and singular families and to pairs $(X,W)$. Central to the method is an obstruction map $\Psi_Y^\Lambda$ on a simple normal crossing special fibre, and a base-change–compatible double-cone degeneration that inductively propagates torsion information from lower-dimensional intersections. The authors show that for a very general degree $d$ hypersurface in characteristic not 2, with dimension up to $N\le (d+1)2^{d-4}$, the diagonal cannot decompose, so the hypersurface is not stably or retract rational nor $\mathbb{A}^1$-connected (with a characteristic-2 analogue). A key ingredient is bounding the torsion order $\operatorname{Tor}(X)$ through degenerations and comparing it to known nontrivial torsion examples, providing new irrationality ranges, including new cases for quintic hypersurfaces. The work thus advances the understanding of rationality for high-dimensional hypersurfaces by refining and unifying previous approaches via a robust obstruction framework.

Abstract

We show that a very general hypersurface of degree d at least 4 and dimension at most $(d+1)2^{d-4}$ over a field of characteristic different from 2 does not admit a decomposition of the diagonal; hence, it is neither stably nor retract rational, nor $\mathbb{A}^1$-connected. Similar results hold in characteristic 2 under a slightly weaker degree bound. This improves earlier results by the second named author and Moe.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Definition 2.4