Birational Models with Irreducible Arc Spaces

Definitions and mathematical statement

The supplied record does not specify a base field, its characteristic, or the precise convention for "variety." The following formulates the question conditionally under the convention that a variety over a field $k$ is an integral separated scheme of finite type over $k$. Whether this convention and which restrictions on $k$ are intended require clarification.

For a $k$-variety $Y$, its arc space $Y_\infty$ is the $k$-scheme representing the functor on commutative $k$-algebras given by \[ A\longmapsto \operatorname{Hom}_k\left(\operatorname{Spec}(A[[t]]),Y\right). \] Irreducibility of $Y_\infty$ means irreducibility of its underlying topological space. A morphism $f:Y\to X$ is birational if it restricts to an isomorphism between dense open subvarieties of $Y$ and $X$. It is proper if it is separated, of finite type, and universally closed.

For a fixed base field $k$ in the intended, but unspecified, class of fields, the question is: does every $k$-variety $X$ admit a $k$-variety $Y$ and a proper birational $k$-morphism $f:Y\to X$ such that $Y_\infty$ is irreducible? Equivalently, is \[ \forall X\quad \exists\,Y\ \exists\,f:Y\to X\quad \left(f\text{ is proper and birational}\right) \ \land\ \left(Y_\infty\text{ is irreducible}\right) \] valid in that setting?

The supplied dated curation labels this question open in general. It describes a positive result concerning a birationally equivalent projective variety under a characteristic bound; it does not supply that bound or a precise theorem statement. This qualification does not replace the requested proper birational morphism to the given $X$, and no such replacement is made here.

Short English statement

Given a variety $X$, ask whether there is a proper birational morphism $Y\to X$ whose source has irreducible arc space. The base field, characteristic restrictions, and convention for varieties remain unspecified. The supplied open status and qualified positive-result assessment require independent review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30004082$, source code catalogue entry, titled "Birational Models with Irreducible Arc Spaces." The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2019; no individual proposer is supplied.
• 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 attributes the problem to workshop open-problem discussions documented in this report; no exact problem page is supplied.
• Supplied literature reference: Benito--Piltant--Reguera, https://doi.org/10.1016/j.jpaa.2022.107110. No article title, precise theorem, or characteristic bound is supplied.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not an independently verified assessment. It labels the problem open in general and describes a positive result only for a birationally equivalent projective variety under an unspecified characteristic bound. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.