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The diagonal of (3,3) fivefolds

Jan Lange, Bjørn Skauli

Abstract

We show that a very general (3,3) complete intersection in $\mathbb{P}^7$ over an algebraically closed uncountable field of characteristic different from 2 admits no decomposition of the diagonal, in particular it is not retract rational. This strengthens Nicaise--Ottem's result, where stable irrationality in characteristic 0 was shown. The main tool is a Chow-theoretic obstruction which was found by Pavic--Schreieder, where quartic fivefolds are studied.

The diagonal of (3,3) fivefolds

Abstract

We show that a very general (3,3) complete intersection in over an algebraically closed uncountable field of characteristic different from 2 admits no decomposition of the diagonal, in particular it is not retract rational. This strengthens Nicaise--Ottem's result, where stable irrationality in characteristic 0 was shown. The main tool is a Chow-theoretic obstruction which was found by Pavic--Schreieder, where quartic fivefolds are studied.
Paper Structure (12 sections, 22 theorems, 94 equations)

This paper contains 12 sections, 22 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and Notations
  4. Decomposition of the diagonal
  5. Chow-theoretic obstruction to retract rationality
  6. Specializations
  7. Alterations
  8. Very general (3,3) fivefolds are irrational
  9. A strictly semi-stable family
  10. Specialization
  11. Proving that $\Phi$ is not surjective
  12. Proof of the main result

Key Result

Theorem 1.1

Let $k$ be an uncountable algebraically closed field of characteristic different from $2$. Then the very general (3,3) fivefold over $k$ admits no decomposition of the diagonal, in particular it is not retract rational.

Theorems & Definitions (44)

  • Theorem 1.1
  • Lemma 2.1
  • Definition 2.2: PS23
  • Lemma 2.3
  • Theorem 2.4: PS23
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Definition 3.1
  • ...and 34 more