no. 3
The Diagonal of (3,3) fivefolds
Lange, Jan1; Skauli, Bjørn2
1Institute of Algebraic Geometry, Leibniz University Hannover, Welfengarten 1, 30167 Hannover (Germany)
2Department of Mathematics, University of Oslo, Moltke Moes vei 35 0851 Oslo (Norway)
Annales de l'Institut Fourier, Volume 76 (2026) no. 3, pp. 1309-1339

We show that a very general (3,3) complete intersection in $\mathbb{P}^7$ over an uncountable algebraically closed field of characteristic different from $2$ 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.

En travaillant sur un corps algébriquement clos non dénombrable de caractéristique différente de 2, nous montrons qu’une intersection complète de bidegré (3,3) et de dimension 5 n’a pas de décomposition de la diagonale. En particulier, une telle intersection complète n’est pas rétracte rationnelle. Cela renforce un résultat de Nicaise et Ottem, qui ont montré qu’en caractéristique 0, une telle intersection complète n’est pas stablement rationnelle. L’outil principal est une obstruction découverte par Pavic et Schreieder, qui ont étudié les hypersurfaces quartiques de dimension cinque.

Received:
Revised:
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Published online:
DOI: 10.5802/aif.3735
Classification: 14M10, 14C25, 14E08
Keywords: complete intersections, rational, retract rational
Mots-clés : intersection complète, rationnelle, rétracte rationnelle

Lange, Jan  1 ; Skauli, Bjørn  2

1 Institute of Algebraic Geometry, Leibniz University Hannover, Welfengarten 1, 30167 Hannover (Germany)
2 Department of Mathematics, University of Oslo, Moltke Moes vei 35 0851 Oslo (Norway)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Lange, Jan; Skauli, Bjørn. The Diagonal of (3,3) fivefolds. Annales de l'Institut Fourier, Volume 76 (2026) no. 3, pp. 1309-1339. doi: 10.5802/aif.3735
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