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Rationality of hypersurfaces

Stefan Schreieder

TL;DR

This survey aggregates state-of-the-art methods in the rationality problem for hypersurfaces, emphasizing cycle-theoretic, unramified cohomology, motivic, and combinatorial (regular matroid) techniques. It explains how diagonal decompositions and CH$_0$-triviality yield concrete obstructions to rationality, and it highlights sharp results: logarithmic and linear degree bounds ruling out retract/stable rationality in broad regimes, and culminates in the irrationality of very general cubic fourfolds and cubic threefolds via intermediate Jacobians and matroid-based obstructions. The work also discusses how modern degeneration and motivic strategies turn rationality questions into computable invariants, and how these interact with families and base fields. Overall, the paper maps a multi-faceted toolkit that has driven substantial progress in understanding when hypersurfaces fail to be rational, with implications for higher-dimensional birational geometry and beyond.

Abstract

We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.

Rationality of hypersurfaces

TL;DR

This survey aggregates state-of-the-art methods in the rationality problem for hypersurfaces, emphasizing cycle-theoretic, unramified cohomology, motivic, and combinatorial (regular matroid) techniques. It explains how diagonal decompositions and CH-triviality yield concrete obstructions to rationality, and it highlights sharp results: logarithmic and linear degree bounds ruling out retract/stable rationality in broad regimes, and culminates in the irrationality of very general cubic fourfolds and cubic threefolds via intermediate Jacobians and matroid-based obstructions. The work also discusses how modern degeneration and motivic strategies turn rationality questions into computable invariants, and how these interact with families and base fields. Overall, the paper maps a multi-faceted toolkit that has driven substantial progress in understanding when hypersurfaces fail to be rational, with implications for higher-dimensional birational geometry and beyond.

Abstract

We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.
Paper Structure (16 sections, 33 theorems, 62 equations)

This paper contains 16 sections, 33 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Examples of rational varieties.
  4. Retract rational varieties.
  5. Functorial birational invariants.
  6. Zero-cycles and decompositions of the diagonal.
  7. Very general hypersurfaces.
  8. Degeneration and unramified cohomology
  9. Degeneration techniques.
  10. Hypersurfaces under a logarithmic degree bound.
  11. Motivic methods and degeneration to unions
  12. Obstructing cohomological decompositions of the diagonal of cubic threefolds
  13. Cohomological decompositions of the diagonal.
  14. Voisin's generalization of the Clemens--Griffiths method.
  15. Regular matroids.
  16. ...and 1 more sections

Key Result

Theorem 1.1

A very general hypersurface $X\subset \mathbb P^{n+1}_k$ of dimension $n\geq 3$ and degree $d\geq (\log_2 n)+2$ over a field of characteristic different from $2$ is not retract rational.

Theorems & Definitions (70)

  • Theorem 1.1: Sch-JAMS
  • Theorem 1.2: NOmoePavic-SchLange-Sch
  • Theorem 1.3: EGFS
  • Theorem 1.4: KKPY
  • Example 2.1: Stereographic projection
  • Example 2.2: Euler 1761
  • Example 2.3: Clebsch 1866
  • Example 2.4
  • Definition 2.5: Saltman saltman-2
  • Example 2.6
  • ...and 60 more