On the motivic approach to the Abhyankar rectifiability problem and concrete rectifications

Abstract

We study closed embeddings of the complex affine line into affine three-space. We prove that, for every such embedding, the complete-intersection normal map identifies the $\mathbb{A}^1$-fundamental sheaf of its complement with $\mathbf{K}_2^{\mathrm{MW}}$. The proof compares torsors under arbitrary strongly $\mathbb{A}^1$-invariant group sheaves. A nodal projection expresses a torsor by a normal parameter and finitely many node parameters. Extension across the node axes gives equations for the latter. Their correction terms increase a nilpotent filtration generated by the Milnor-Witt symbols of the branch-height differences. We deduce the vanishing of the first Alexander sheaf of every complex embedded line. We also construct polynomial rectifications using finite inverses and affine modifications, including rectifications of the Shastri trefoil and a harmonic figure-eight over $\mathbb{Z}[a]/(a^2-a+1)$, with arbitrary base change. The companion repository (https://github.com/qft98romanov-hash/abhyankar-moh-trefoil) contains explicit rectifiers, their inverses, proofs, and symbolic checks. We prove that a normal map with a polynomial section and geometric generic affine-line fiber is a coordinate projection. We formulate the remaining rectifiability and descent questions; the general complex rectifiability problem is not settled here. An appendix treats other fields, normal orientations, and arithmetic descent.