# The diagonal of (3,3) fivefolds

## TL;DR

This paper proves that very general (3,3) complete intersections in projective 7-space are not retract rational by using Chow-theoretic obstructions, extending previous results on stable irrationality.

## Contribution

It introduces a new Chow-theoretic obstruction to show that certain (3,3) fivefolds are not retract rational, strengthening prior stable irrationality results.

## Key findings

- Very general (3,3) fivefolds are not retract rational.
- The Chow-theoretic obstruction effectively detects non-rationality.
- Extension of irrationality results to positive characteristic fields.

## 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.