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On the diagonal of low bidegree hypersurfaces

Morten Lüders, Elia Fiammengo

TL;DR

The paper develops a cycle-theoretic obstruction to the decomposition of the diagonal for bidegree hypersurfaces in products of projective spaces, using Lange–Schreieder degenerations to propagate obstructions to higher degree and dimension. Starting from Hassett–Pirutka–Tschinkel’s quadric surface bundle and very general $(2,2)$ hypersurfaces, it proves a general raising-degree-and-dimension principle and a dimension-raising variant, enabling nonexistence of diagonal decompositions for broader families. It applies the framework to show that a very general $(3,2)$ hypersurface in $\mathbb P^4_k\times\mathbb P^3_k$ does not admit a decomposition of the diagonal over fields with characteristic not equal to two, hence is not retract rational, and derives several further corollaries including conditional reductions to cubic cases and results for Küchle varieties of type $b_4$. The methods illuminate how degeneration techniques yield strong obstructions to diagonal decompositions across multiple bidegree regimes, enriching the landscape of (stable/retract) rationality via diagonal obstructions.

Abstract

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure that allows one to raise the degree and dimension starting from the quadric surface bundle of Hassett, Pirutka and Tschinkel. Furthermore, we are able to raise the dimension without raising the degree in a special case, showing that a very general $(3,2)$ complete intersection in $\mathbb P^4\times \mathbb P^3$ does not admit a decomposition of the diagonal. As a corollary of these theorems, we show that in a certain range, bidegree hypersurfaces which were previously only known to be stably irrational over fields of characteristic zero by results of Moe, Nicaise and Ottem, are not retract rational over fields of characteristic different from two.

On the diagonal of low bidegree hypersurfaces

TL;DR

The paper develops a cycle-theoretic obstruction to the decomposition of the diagonal for bidegree hypersurfaces in products of projective spaces, using Lange–Schreieder degenerations to propagate obstructions to higher degree and dimension. Starting from Hassett–Pirutka–Tschinkel’s quadric surface bundle and very general hypersurfaces, it proves a general raising-degree-and-dimension principle and a dimension-raising variant, enabling nonexistence of diagonal decompositions for broader families. It applies the framework to show that a very general hypersurface in does not admit a decomposition of the diagonal over fields with characteristic not equal to two, hence is not retract rational, and derives several further corollaries including conditional reductions to cubic cases and results for Küchle varieties of type . The methods illuminate how degeneration techniques yield strong obstructions to diagonal decompositions across multiple bidegree regimes, enriching the landscape of (stable/retract) rationality via diagonal obstructions.

Abstract

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure that allows one to raise the degree and dimension starting from the quadric surface bundle of Hassett, Pirutka and Tschinkel. Furthermore, we are able to raise the dimension without raising the degree in a special case, showing that a very general complete intersection in does not admit a decomposition of the diagonal. As a corollary of these theorems, we show that in a certain range, bidegree hypersurfaces which were previously only known to be stably irrational over fields of characteristic zero by results of Moe, Nicaise and Ottem, are not retract rational over fields of characteristic different from two.
Paper Structure (12 sections, 32 theorems, 126 equations, 2 tables)

This paper contains 12 sections, 32 theorems, 126 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Obstruction to the relative decomposition of the diagonal
  4. The relative torsion order
  5. Torsion order relative to $CH_0$-bounded and universally $CH_0$-trivial varieties
  6. Unramified cohomology classes and relative torsion order
  7. Very general bidegree hypersurfaces
  8. Fourfolds of bidegree $(2,3)$ in $\mathbb P^1\times \mathbb P^4$
  9. Küchle varieties of type $b_4$
  10. Raising dimension and degree
  11. Raising dimension without raising degree:Sixfolds of bidegree $(3,2)$ in $\mathbb P^4\times \mathbb P^3$
  12. Applications

Key Result

Theorem 1.1

Let $k$ be a field of characteristic different from two and $\Lambda$ a ring of positive characteristic such that the exponential characterristic of $k$ is invertible in $\Lambda$. Let $n\geq r+1\geq 1$ and assume there exists an integral bidegree $(d,f)$ hypersurface such that $\mathop{\mathrm{Tor}}\nolimits^{\Lambda}(X,\{y_{0}x_{0}y_{1}x_{1}=0\})\neq 1$. Then for any integers $n'\geq n$, $f'\ge

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.3
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • ...and 72 more