The Diagonal of (3,3) fivefolds

We show that a very general (3,3) complete intersection in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℙ</mml:mi> <mml:mn>7</mml:mn> </mml:msup> </mml:math> over an uncountable algebraically closed field of characteristic different from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> admits no decomposition of the diagonal, in particular it is not retract rational. This strengthens Nicaise and Ottem’s result proving stable irrationality of this complete intersection in characteristic 0. The main tool is a Chow-theoretic obstruction which was found by Pavic and Schreieder, who studied quartic fivefolds.