Sharpness of Irrationality after One Stabilization for Cubic Threefolds

Abstract

The irrationality of X times P^1 for every smooth complex cubic threefold X is sharp: two explicit smooth cubic threefolds over Q are rational after multiplication by P^2. For each example, the least m for which X times P^m is rational is exactly two, over every characteristic-zero ground field. We prove more generally that if k has characteristic zero and S/k is a smooth quartic del Pezzo surface that has a rational point and whose geometric Picard lattice is stably permutation, then S times A^2, equivalently S times P^2, is k-rational. Applying this theorem to both Tschinkel–Zhang series of smooth cubic hypersurfaces gives a uniform P^2 rationality bound. Consequently there are smooth projective nonrational fourfolds Y over Q for which Y times A^1 is rational, answering the affine-line stabilization question raised there. The release includes an exact-arithmetic certificate, identities certifying the six empty localized cases, and generated manuscript values for the four-open tangent-section calculation; the new rationality theorem is not formalized in Lean.