What does this research mean for the field?

The irrationality of X times P^1 for smooth complex cubic threefolds is sharp, as demonstrated by explicit examples that become rational after product with P^2, thereby answering the affine-line stabilization question. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine the minimal stabilization dimension required for smooth cubic threefolds to achieve rationality over rational and complex fields.

August 27, 2026Open Access

Sharpness of Irrationality after One Stabilization for Cubic Threefolds

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Authors

TRTavis Rudd

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Implication

Theoretical analysis demonstrates sharp rationality bounds for stabilized cubic threefolds, resolving long-standing questions regarding affine-line stabilization.

Key Points

To determine the minimal stabilization dimension required for smooth cubic threefolds to achieve rationality over rational and complex fields.
Constructed explicit smooth cubic threefolds over the rational field Q and analyzed their rationality under projective space products.
Characterized smooth quartic del Pezzo surfaces in characteristic zero possessing a rational point and a stably permutation geometric Picard lattice.
Applied theoretical frameworks to the Tschinkel–Zhang series of smooth cubic hypersurfaces supported by exact-arithmetic certificates.
Established that the minimal integer m for which the product of the cubic threefold X with P^m is rational is strictly m = 2 over both Q and C.
Proved that smooth quartic del Pezzo surfaces with a rational point and stably permutation geometric Picard lattice become k-rational when multiplied by A^2 or P^2.
Confirmed the existence of smooth projective nonrational fourfolds Y over Q where Y times A^1 is rational, resolving the affine-line stabilization problem.

Cite This Study

Tavis Rudd (2026) studied this question.

synapsesocial.com/papers/6a8fe9da10c91c1e92621cd4 https://doi.org/10.5281/zenodo.22089294

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Implication

Theoretical analysis demonstrates sharp rationality bounds for stabilized cubic threefolds, resolving long-standing questions regarding affine-line stabilization.

Sharpness of Irrationality after One Stabilization for Cubic Threefolds

Discussion

Member takes

Implication

Key Points

Cite This Study

Sharpness of Irrationality after One Stabilization for Cubic Threefolds

Discussion

Member takes

Implication

Key Points

Cite This Study

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