Rigidity of Morphisms to Wound Unipotent Groups

Definitions and mathematical statement

Let $k$ be a field, with no restriction on its characteristic. A unipotent algebraic $k$-group $U$ is called $k$-wound if every $k$-morphism $\mathbb A_k^1\to U$ is constant. The supplied record does not specify its smoothness and connectedness conventions for unipotent groups; the intended convention requires confirmation.

Write $e:\operatorname{Spec}k\to U$ for the identity section. For any $k$-scheme $T$, let $1_T:T\to U$ denote the composite of its structure morphism with $e$. All products below are over $k$.

The proposed rigidity assertion is the following. For every field $k$, every wound unipotent group $U$ over $k$, every closed point $x\in\mathbb P_k^1$, and every pair $y_1,y_2\in X(k)$, where \[ X=\mathbb P_k^1\setminus\{x\}, \] every $k$-morphism $f:X\times_k X\to U$ satisfying \[ f\circ i_1=1_X,\qquad f\circ i_2=1_X, \] where $i_1(t)=(y_1,t)$ and $i_2(t)=(t,y_2)$, satisfies \[ f=1_{X\times_k X}. \] These are equalities of scheme morphisms, rather than merely equalities on rational points. The morphism $f$ is not assumed to be a group homomorphism. No distinctness condition on $y_1,y_2$, degree restriction on $x$, or separability hypothesis on its residue field is supplied.

The record presents this as a conjecture and labels it partially solved. Its supplied dated assessment says that the assertion is proved in characteristic different from two, with structural applications there, while the characteristic-two case and the full all-characteristic assertion remain unresolved. This assessment is not independently verified here; the structural applications are not specified in the record.

Short English statement

The conjecture asks whether a morphism from the product of two copies of a punctured projective line to a wound unipotent group must be constantly the identity whenever it is the identity along both specified coordinate slices. The supplied assessment reports a proof when $\operatorname{char}(k)\ne 2$, with characteristic two unresolved. The group conventions and dated literature claims require human review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004661$, titled Rigidity of Morphisms to Wound Unipotent Groups; source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2021; no individual proposer is named.
• Supplied source citation: Algebraic Groups (2022), Oberwolfach Reports, pp. 1087--1148, DOI 10.4171/owr/2021/20. https://doi.org/10.4171/owr/2021/20. The literature-source list calls this Oberwolfach Report 20/2021. The discrepancy between the supplied title year and report identifier requires verification.
• Supplied literature reference: Achet, Unirational Algebraic Groups. https://hal.science/hal-02358528/document.
• Literature assessment checked at 2026-08-22: supplied dated curation, not independently verified. It reports a proof in characteristic different from two and structural applications there, with characteristic two and the full all-characteristic assertion unresolved. The background explicitly requests independent verification of time-sensitive status and cited claims.