Higher Product Rigidity for Wound Unipotent Targets

Definitions and mathematical statement

Let $k$ be a field, let $n\geq 1$ be an integer, and let $U$ be a wound unipotent group over $k$, as in the supplied record. No restriction on $\operatorname{char}(k)$ is supplied. Under the usual smooth connected convention, $k$-wound means that every $k$-morphism $\mathbb A^1_k\to U$ is constant; the precise conventions for the group hypotheses should be checked against the cited report.

For each $i\in\{1,\ldots,n\}$, let $X_i\subseteq\mathbb P^1_k$ be a nonempty open subscheme and choose $x_i\in X_i(k)$. Write $D$ for the source's union of their complementary divisors, whose support is \[ \bigcup_{i=1}^{n}\left(\mathbb P^1_k\setminus X_i\right). \] The record does not specify divisor multiplicities or define the degree of primitivity of $D$. Denote that unspecified invariant provisionally by $\delta(D)$. This notation supplies no definition of the invariant, and its value when $D$ is empty is also unspecified.

Set $P=\prod_{j=1}^{n}X_j$, with all products taken over $k$. For each $i$, let \[ \iota_i:\prod_{j\ne i}X_j\longrightarrow P \] be the morphism inserting $x_i$ in the $i$-th coordinate. An empty product here means $\operatorname{Spec}(k)$. For a $k$-scheme $T$, write $1_T:T\to U$ for the composite of its structure morphism with the identity section of $U$.

The supplied conjecture asks whether, for every such collection of data, \[ n>\delta(D) \quad\Longrightarrow\quad \left[ \forall f\in\operatorname{Hom}_k(P,U),\quad \left(\forall i\in\{1,\ldots,n\},\ f\circ\iota_i=1_{\prod_{j\ne i}X_j}\right) \Longrightarrow f=1_P \right]. \] The face restrictions and conclusion are equalities of morphisms of $k$-schemes, not merely equalities on $k$-rational points. This is a provisional formulation: the missing definition of $\delta(D)$ prevents a fully specified mathematical assertion.

Short English statement

The conjecture says that a morphism from a product of pointed open subsets of $\mathbb P^1_k$ to a wound unipotent group must be constantly the identity if it is the identity on every chosen coordinate face and the number of factors exceeds the degree of primitivity of the boundary. The supplied record leaves that boundary invariant undefined, so the numerical hypothesis requires clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004662$, exact source problem code catalogue entry, title Higher Product Rigidity for Wound Unipotent Targets, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2021 and names no individual proposer.
• Supplied source citation: Algebraic Groups (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/20, pp. 1087--1148. https://doi.org/10.4171/owr/2021/20. The literature-source list labels this same reference Oberwolfach Report 20/2021. Both year designations are retained as supplied; their relationship has not been independently checked.
• Supplied literature source: Achet, Unirational Algebraic Groups. https://hal.science/hal-02358528/document.
• Literature assessment dated exactly 2026-08-22: this is supplied dated curation, not an independently verified assessment. The record labels the problem open and describes the assertion as a conjectural strengthening of a "characteristic-one case." It reports that no proof of the general assertion or its descending-central-series consequence was found. Neither the comparison case nor that consequence is formulated in the supplied record. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports manual semantic, typographic, and LaTeX review claiming that the clean statement is self-contained and preserves the target without unsupported additions. These are supplied review claims; they do not resolve the missing definition identified in this draft.