On the diagonal of quartic hypersurfaces and (2,3)-complete intersection n-folds

2025/10/08 by Elia Fiammengo, Fiammengo, Elia, Morten Lüders +1 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2510.07111

openalex publication_date 2025/10/08 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We study the question of the existence of a decomposition of the diagonal for very general quartic and (2,3)-complete intersection n-folds. Using cycle-theoretic techniques of Lange, Pavic and Schreieder we reduce the question via a degeneration argument to the existence of such a decomposition for n-1-dimensional cubic hypersurfaces and their essential dimension. A result of Voisin on the essential dimension of complex cubic hypersurfaces of odd dimension (and of dimension four) then yields conditional statements that extend results of Nicaise and Ottem from stable rationality to the existence of a decomposition of the diagonal. As an application, we use a recent result of Engel, de Gaay Fortman and Schreieder on the decomposition of the diagonal for cubic threefolds to give a new proof of the non-retract rationality of a very general complex quartic 4-fold, originally due to Totaro, and of a very general complex (2,3)-complete intersection 4-fold, originally due to Skauli.

Citations

• Torsion order and irrationality of complete intersections
• Matroids and the integral Hodge conjecture for abelian varieties
• On the rationality problem for low degree hypersurfaces
• On Stable Rationality of Polytopes
• The diagonal of (3,3) fivefolds
• The diagonal of quartic fivefolds
• Unramified cohomology, algebraic cycles and rationality
• A refinement of the motivic volume, and specialization of birational types
• Prelog Chow groups of self-products of degenerations of cubic threefolds
• Tropical degenerations and stable rationality
• Specialization of birational types
• The motivic nearby fiber and degeneration of stable rationality
• Remarks on approximate decompositions of the diagonal
• Stable rationality of quadric surface bundles over surfaces
• Hypersurfaces that are not stably rational
• Universal unramified cohomology of cubic fourfolds containing a plane

Cited by

• Torsion order and irrationality of complete intersections

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• The diagonal of quartic fivefolds
• The diagonal of (3,3) fivefolds
• A Stably Irrational (2,3)-Complete Intersection Fourfold over ℚ
• A classification of terminal quartic 3-folds and applications to rationality questions
• Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi-Yau complete intersections