Finiteness of Singular-Centered Arc-Space Components

Definitions and mathematical statement

Let $p$ be a prime, let $k$ be a perfect field of characteristic $p>0$, and let $X$ be an irreducible variety over $k$. Here the usual convention is used that a variety is a reduced, separated scheme of finite type over its ground field; geometric irreducibility is not assumed. There is no restriction on $\dim X$.

The arc space $X_\infty$ is the $k$-scheme representing the functor that assigns to a $k$-algebra $A$ the set \[ \operatorname{Hom}_k\left(\operatorname{Spec} A[[t]],X\right). \] Evaluation at $t=0$ defines the center morphism $\pi_0:X_\infty\to X$. Write $\operatorname{Sing}(X)$ for the locus where $X$ is not regular, equipped with its reduced closed subscheme structure. Since $k$ is perfect, this is also the locus where $X\to\operatorname{Spec}k$ is not smooth. Define \[ S_X:=X_\infty\times_X\operatorname{Sing}(X). \] For a field extension $K/k$, an arc $\gamma:\operatorname{Spec}K[[t]]\to X$ has center $\gamma|_{\operatorname{Spec}K}$, and belongs to $S_X$ precisely when that center lies in $\operatorname{Sing}(X)$.

Question. For every such $k$ and $X$, does the underlying Zariski topological space of $S_X$ have finitely many irreducible components?

The condition concerns the center of an arc; it does not require the whole arc to factor through $\operatorname{Sing}(X)$. Components are those of the scheme $S_X$, rather than components of a set of $k$-rational arcs or of an unspecified geometric base change. If $X$ is smooth, then $S_X=\varnothing$, which has zero irreducible components. The source does not explicitly state its convention for the word "variety"; the convention above requires confirmation against the cited report.

Short English statement

For every irreducible variety over a perfect field of positive characteristic, ask whether the space of all formal arcs whose centers lie in the singular locus has finitely many irreducible components. The question has no dimension bound and allows arcs whose generic points lie outside the singular locus. The supplied dated curation labels the general problem open; that assessment is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004081$, source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the recorded proposed year is 2019.
• Supplied report citation: Singularities and Homological Aspects of Commutative Algebra (2020), Oberwolfach Reports, pp. 407--464, DOI 10.4171/owr/2019/6. https://doi.org/10.4171/owr/2019/6. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Benito--Piltant--Reguera. https://doi.org/10.1016/j.jpaa.2022.107110. No article title or specific supporting result is supplied.
• Supplied literature reference: Piltant--Reguera. https://doi.org/10.1016/j.jpaa.2017.08.012. No article title or specific supporting result is supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that finiteness in arbitrary dimension in positive characteristic remains unknown, while surface and local-uniformization cases do not establish the general statement. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Time-sensitive status and the cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified, explaining that mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired. This is a supplied editorial characterization, not mathematical verification of this draft.