On the diagonal of low bidegree hypersurfaces

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure that allows one to raise the degree and dimension starting from the quadric surface bundle of Hassett, Pirutka and Tschinkel. Furthermore, we are able to raise the dimension without raising the degree in a special case, showing that a very general $(3,2)$ complete intersection in $\mathbb P^4\times \mathbb P^3$ does not admit a decomposition of the diagonal. As a corollary of these theorems, we show that in a certain range, bidegree hypersurfaces which were previously only known to be stably irrational over fields of characteristic zero by results of Moe, Nicaise and Ottem, are not retract rational over fields of characteristic different from two.

💡 Research Summary

The paper investigates the existence of a decomposition of the diagonal for hypersurfaces of low bidegree in a product of projective spaces, a property that is equivalent to the variety being rational, stably rational, or retract rational. The authors introduce the notion of torsion order, an invariant measuring the minimal integer e for which a multiple e·Δ of the diagonal can be expressed as a sum of a product of a zero‑cycle and the whole variety plus a cycle supported away from the second factor. They also define a relative version of this invariant and relate it to CH₀‑boundedness and universal CH₀‑triviality.

The technical core relies on a cycle‑theoretic degeneration method due to Lange, Pavic and Schreieder. By constructing a strictly semistable degeneration of a given hypersurface, one can detect non‑trivial torsion order on lower‑dimensional strata, which then propagates to the original variety. Using this tool, the authors develop two inductive “raising” procedures.

The first procedure raises both the dimension and the degree simultaneously. Starting from a known example— the quadric surface bundle Q studied by Hassett, Pirutka and Tschinkel— they consider a (d,f)‑(2,0) complete intersection in a larger product space. This complete intersection is birational to a hypersurface of bidegree (d+1,f) in a product with one fewer factor of the first projective space. By iterating this construction, they prove Theorem 1.1: if a very general (d,f) hypersurface in Pⁿ⁻ʳ×Pʳ⁺¹ has torsion order 1 (i.e. admits a decomposition of the diagonal), then any very general hypersurface with larger n′, larger f′ and degree d′≥d+(n′−n) also has torsion order 1. Conversely, starting from a hypersurface known not to admit a decomposition, the same argument shows that all such “raised” hypersurfaces also fail to admit one.

The second, more delicate, procedure raises only the dimension while keeping the bidegree fixed. The authors apply it to the (3,2) case. Using the same degeneration strategy on Q, they first obtain a (3,2)‑(2,0) complete intersection in P⁵×P³, then show that this complete intersection is birational to a (3,2) hypersurface in P⁴×P³. Theorem 1.2 establishes that for a very general (3,2) hypersurface in P⁴×P³ over a field of characteristic ≠ 2, the torsion order with respect to the cycle {y₀x₀ y₁x₁ y₂x₂ = 0} is not 1, i.e. the diagonal cannot be decomposed. This result fills a gap left by earlier work of Moë, which only proved stable irrationality for such hypersurfaces.

Combining these theorems, the authors obtain a broad range of new non‑rationality statements (Theorem 1.3). For instance, very general hypersurfaces of bidegree (d,2) with d≥n−2 in Pⁿ⁻¹×P², of bidegree (d,2) with d≥n−3 in Pⁿ⁻²×P³, and of bidegree (d,3) with d≥n−3 in Pⁿ⁻²×P³ all fail to admit a decomposition of the diagonal, and hence are not retract rational over fields of characteristic different from two. Corollary 1.4 shows that, assuming the non‑existence of a diagonal decomposition for very general cubic hypersurfaces in Pʳ⁺¹ (with r odd or r=4), the same holds for a wide family of bidegree (d,3) hypersurfaces. Corollary 1.5 proves a similar statement for very general (2,4) hypersurfaces in P¹×P⁶.

The paper also situates its results within the broader landscape of rationality problems for quadric bundles, referencing the optimality result of Schreieder (Theorem 1.7) and explaining how the new bounds improve upon earlier stable‑irrationality results of Nicaise–Ottem and others. Throughout, the authors emphasize that their approach—raising dimensions while keeping degrees low—yields stronger conclusions in low‑degree cases than previous methods that typically raise both degree and dimension simultaneously.

In summary, by blending the Lange–Pavic–Schreieder degeneration technique with careful birational constructions, the authors provide a powerful inductive framework that produces many new examples of hypersurfaces of low bidegree which do not admit a decomposition of the diagonal, thereby establishing non‑retract‑rationality over a wide class of fields. This advances the understanding of rationality phenomena for multidegree hypersurfaces and opens avenues for further exploration of torsion order as a diagnostic tool.