Bad Hyperplanes and Residual Variables in Polynomial Rings

Definitions and mathematical statement

Let $d\geq3$, let $R$ be the characteristic-zero ring intended in the source, and put $A=R[x_1,\ldots,x_d]$. In the usual commutative unital convention, characteristic zero does not imply that $R$ is a field, a domain, or a ${\mathbb{Q}}$-algebra, nor that its residue fields have characteristic zero. No such stronger assumption is supplied.

A coordinate of $A$ over $R$ is an element $f$ extendable to polynomial coordinates, so that $A=R[f,g_2,\ldots,g_d]$ as a polynomial algebra. A usual residual-variable convention requires that for every prime $\mathfrak p\subset R$, the image $\overline f$ be a coordinate of $\kappa(\mathfrak p)[x_1,\ldots,x_d]$. A "bad residual variable" might mean one satisfying this fiberwise property but not the global coordinate property; the source does not reproduce its definition and that interpretation must be checked. Merely requiring each fiber quotient by $\overline f$ to be affine space is a different condition and is not silently identified with fiberwise coordinateness.

Similarly, an algebraic hyperplane may mean a hypersurface $V(f)\subset\operatorname{Spec}A$ isomorphic over $R$ to ${\mathbb{A}}_R^{d-1}$, equivalently $A/(f)\cong R^{[d-1]}$. Calling it bad could refer to failure of rectification to a coordinate hyperplane, noncoordinateness of $f$, or a stronger source-specific property. It need not mean a linear degree-one hyperplane. The corresponding bad-line constructions and their precise defect are not supplied.

The source task is to generalize those constructions to obtain bad hyperplanes and bad residual variables in $d$-variable polynomial rings over the intended characteristic-zero bases. It does not give a construction to extend, define bad, state whether every such $R$ is intended or only suitable examples, or prescribe the additional properties to be preserved. Therefore neither a universal existence theorem for all characteristic-zero rings nor an invented higher-dimensional example is asserted.

The supplied literature triage checked 2026-08-21 reports no authoritative complete resolution of the extracted generalization request, without separate definitions or theorem citations.

Short English statement

Extend the source's bad-line and residual-variable constructions to $d\geq3$. The meaning of bad, the original construction, and the allowed base rings are missing; fiberwise coordinates, affine-space quotients, and global rectifiability must remain distinct.

Sources

• Local UnsolvedMath registry, permanent ID 30000642, catalogue entry, Bad Hyperplanes and Residual Variables in Polynomial Rings.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI https://doi.org/10.4171/owr/2007/01.
• Supplied literature triage checked 2026-08-21 cites only the original report; no bad-line construction, formal residual-variable definition, or separate theorem reference is supplied.