Nontame Automorphisms with Point-Centered Sarkisov Factorizations

Definitions and mathematical statement

Use the standard compactification ${\mathbb{C}}^3\subset{\mathbb{P}}^3_{\mathbb{C}}$ with hyperplane at infinity $H_\infty={\mathbb{P}}^3_{\mathbb{C}}\setminus{\mathbb{C}}^3$. A polynomial automorphism $\theta:{\mathbb{C}}^3\to{\mathbb{C}}^3$ has a polynomial inverse and induces a birational, or Cremona, transformation \[ \Phi_\theta:{\mathbb{P}}^3_{\mathbb{C}}\dashrightarrow{\mathbb{P}}^3_{\mathbb{C}} \] whose restriction to the affine chart is $\theta$.

The source asks whether there exists an essentially nontame such $\theta$ whose Sarkisov factorization begins with a blowup, possibly weighted, centered at a point $p\in H_\infty$. Thus all of the following source qualifications are required: polynomial invertibility on affine three-space over ${\mathbb{C}}$, essential nontameness, the induced projective transformation, and an initial point-centered extraction at infinity, allowing weights.

For context, the tame subgroup of polynomial automorphisms is ordinarily generated by affine automorphisms and triangular automorphisms. Nontame means outside that subgroup. The stronger phrase "essentially nontame" is not defined in the record, so it is not replaced by ordinary nontameness or by an arbitrary choice of conjugation or composition equivalence.

A Sarkisov factorization expresses a birational map through the relevant birational links of Mori fiber spaces. The exact link convention, allowed singularities and weights, and meaning of the initial blowup must be recovered from the report. Such factorizations need not be unique, so the phrase "the Sarkisov factorization" leaves open whether the requested automorphism should admit at least one factorization with that first extraction, satisfy it for a designated procedure, or satisfy a stronger condition for every factorization. The source gives no selection procedure. No curve-centered extraction or blowup away from $H_\infty$ is substituted for the stipulated point center.

Short English statement

The existence question seeks an essentially nontame polynomial automorphism of ${\mathbb{C}}^3$ whose projective Sarkisov factorization starts by blowing up a point at infinity, possibly with weights. Essential nontameness and the choice of factorization are undefined in the extraction.

Sources

• Local UnsolvedMath registry, permanent ID 30000640, code catalogue entry, "Nontame Automorphisms with Point-Centered Sarkisov Factorizations." Proposed year 2007; proposer unspecified.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI 10.4171/owr/2007/01; https://doi.org/10.4171/owr/2007/01. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says automorphism and affine/projective-space notation was enclosed in balanced delimiters, retaining every geometric qualifier.
• Literature triage checked 2026-08-21: open. It generically describes the item as an unresolved problem/conjecture/direction and says no authoritative complete resolution was found by the supplied targeted check. It does not define essential nontameness or the factorization choice. Created 2026-08-25; updated 2026-09-16. Current status and source scope require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.