Contact Loci and Nash Sets of Prime Divisors

Definitions and mathematical statement

A prime divisor is an irreducible codimension-one subvariety of a model. The supplied record considers a prime divisor $E$, its minimal model, and a free infinitely near point $p_{i_0}$ used to obtain that model. It does not specify the ambient space, the base field, the construction of the minimal model, or the convention for a free infinitely near point.

The notation $E_{p_{i_0}}$ denotes the divisor associated in the source with $p_{i_0}$; its construction is not supplied. The notation $\mathcal N_{E'}$ denotes the Nash set associated with a prime divisor $E'$, and $E'\equiv_{i_0}E$ is the source's relation indexing the divisors in the union. Neither the Nash-set convention nor this relation is defined in the supplied record. Likewise, $\operatorname{Cont}_q(E_{p_{i_0}})$ denotes the source's contact locus, whose ambient arc space and contact-order convention are unspecified.

For a prime divisor $E$ and a free infinitely near point $p_{i_0}$ used to obtain its minimal model, the question is whether there exists a contact-order parameter $q$ such that \[ \bigcup_{E'\equiv_{i_0}E}\mathcal N_{E'} = \operatorname{Cont}_q\left(E_{p_{i_0}}\right). \] The parameter $q$ is existentially quantified for the given data; its allowed domain is not stated. The left-hand side is the union displayed in the source, with no additional closure. This is a schematic transcription of the proposed equality: the missing definitions and ambient identifications must be recovered before it constitutes a fully specified mathematical statement.

Short English statement

Given a prime divisor $E$ and a free infinitely near point $p_{i_0}$ in the construction of its minimal model, ask whether the union of the Nash sets indexed by $E'\equiv_{i_0}E$ is a single contact locus of $E_{p_{i_0}}$, for some order $q$. The supplied record leaves essential notation and ambient assumptions undefined.

Sources

• Local UnsolvedMath registry, permanent ID $30003225$, title "Contact Loci and Nash Sets of Prime Divisors," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified in the supplied record; the proposed year is recorded as 2016.
• Supplied report citation: Singularities (2017), Oberwolfach Reports, pp. 2625--2691, DOI 10.4171/owr/2016/46. https://doi.org/10.4171/owr/2016/46. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's 2017 label is retained alongside the record's 2016 proposed year and report identifier.
• Supplied literature source: Fernandez de Bobadilla--Pere Pereira--Popescu-Pampu, "generalized Nash problem." https://doi.org/10.1016/j.aim.2017.09.035.
• Supplied literature source: "Oberwolfach Report 46." https://doi.org/10.4171/owr/2016/46.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the generalized Nash-adjacency problem has been reduced to combinatorics, that its relation to plane-curve adjacencies has been clarified, and that the specific contact-locus equality in this record is not proved in general. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The supplied background requests independent verification of time-sensitive status and cited claims.