Generic Splitting Varieties for Pointed Functors

Definitions and mathematical statement

Let $k$ be a field and let $S$ be a functor from field extensions of $k$ to pointed sets. For each extension $l/k$, denote the distinguished point of $S(l)$ by $*_l$. For $s\in S(k)$, write $s_l$ for its image under the restriction map $S(k)\to S(l)$; thus $s_l$ is trivial precisely when $s_l=*_l$.

A $k$-variety is understood here in the usual sense of a separated scheme of finite type over $k$. It is geometrically integral if its base change to an algebraic closure of $k$ is integral. For a $k$-variety $X$ and an extension $l/k$, put $X_l=X\times_{\operatorname{Spec}k}\operatorname{Spec}l$ and $X(l)=\operatorname{Hom}_k(\operatorname{Spec}l,X)$.

Question as supplied. For every field $k$, every functor $S$ as above, and every $s\in S(k)$, does there exist a smooth geometrically integral $k$-variety $X$ such that, simultaneously for every field extension $l/k$, \[ s_l=*_l\quad\Longleftrightarrow\quad X(l)\ne\varnothing, \] and such that $X(l)$ is Zariski-dense whenever it is nonempty? The variety $X$ may depend on $(k,S,s)$, but must be chosen independently of $l$.

Unresolved qualifications. The source says that $X(l)$ is dense "in $X$." Taken literally, this means that the images in the underlying topological space of $X$ of all morphisms in $X(l)$ have closure $X$. Another possible intended condition is that the corresponding $l$-rational points are Zariski-dense in $X_l$. The supplied record does not distinguish these formulations, so the ambient space for the density requirement requires confirmation. No additional hypotheses on $S$, no restriction to infinite fields, and no restriction on the extensions $l/k$ are supplied. In particular, the quantifiers include finite fields and the extension $k/k$. Whether the unrestricted functor formulation captures the intended workshop problem requires source review.

Short English statement

For a field $k$, a pointed functor $S$, and an element $s\in S(k)$, the question asks for one smooth geometrically integral variety whose rational points over every extension detect exactly when $s$ becomes trivial. Whenever such points exist, they must also be Zariski-dense. The source leaves the ambient space for density and any intended restrictions on the functor unresolved. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003803$, title Generic Splitting Varieties for Pointed Functors, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. No individual proposer is specified; the supplied proposed year is 2018.
• Supplied report citation: Quadratic Forms and Related Structures over Fields (2019), Oberwolfach Reports, pp. 1235--1291, DOI 10.4171/owr/2018/21. https://doi.org/10.4171/owr/2018/21. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports. The report title's year 2019 and the supplied year 2018 are retained without reconciliation.
• Supplied literature reference labelled "Demarche and Florence on norm varieties": https://arxiv.org/abs/1711.06585.
• Supplied literature reference labelled "Rost norm varieties": https://doi.org/10.1016/j.jpaa.2005.12.012.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that Rost norm varieties give the required criterion only up to a zero cycle of degree prime to the symbol prime, while the general one-variety rational-point criterion remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying note reports manual semantic, typographic, and LaTeX review. These are registry assertions and do not constitute independent mathematical verification of this draft.