Indeterminacy Locus of the Perfect Cone Prym Map

Definitions and mathematical statement

Write $P_g^{\mathrm{Perf}}\colon X_g\dashrightarrow Y_g^{\mathrm{Perf}}$ for the rational extension of the Prym map mentioned in the supplied record, where $X_g$ denotes its intended compactified source and $Y_g^{\mathrm{Perf}}$ its intended perfect-cone target. These symbols are placeholders: the record does not specify the source compactification, the target dimension, the base field, or the admissible values of $g$.

For specified source and target spaces, let $U_g\subseteq X_g$ be the locus of points having an open neighbourhood on which this rational map extends to a regular morphism. Its indeterminacy locus is \[ \operatorname{Indet}(P_g^{\mathrm{Perf}})=X_g\setminus U_g. \] The symbols $FS_2$ and $FS_3$ refer to the two Friedman--Smith loci named in the supplied assessment. Their precise definitions, ambient spaces, and closure conventions are not supplied.

The question recorded by the source is whether \[ \operatorname{Indet}(P_g^{\mathrm{Perf}})=FS_2\cup FS_3. \] This formula preserves the stated target question, but a fully specified mathematical assertion requires definitions of the moduli spaces and Friedman--Smith loci, a range and quantifier for $g$, and clarification of whether equality concerns underlying sets, reduced closed loci, or scheme structures. The supplied assessment mentions higher genus without specifying a numerical range.

Short English statement

Does the rational extension of the Prym map to the perfect-cone compactification fail to be regular precisely along the union of the Friedman--Smith loci $FS_2$ and $FS_3$? The source leaves the moduli conventions, genus range, and precise meanings of these loci unresolved.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002653$, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2014.
• Supplied report citation: Komplexe Analysis (2015), Oberwolfach Reports, pp. 2177--2234, DOI 10.4171/owr/2014/39. https://doi.org/10.4171/owr/2014/39. This is also the supplied literature link labelled "original Oberwolfach report." The citation's 2015 year and the record's 2014 attribution are retained without reconciliation.
• Supplied literature reference: Casalaina-Martin et al., "Prym compactifications," DOI 10.4171/JEMS/678. https://doi.org/10.4171/JEMS/678. No more detailed bibliographic information or pinpoint reference is supplied.
• Literature assessment dated exactly 2026-08-21: the record labels the problem open and states that established inclusions bound the perfect-cone Prym indeterminacy locus in general, while equality with the two Friedman--Smith loci remains unresolved in higher genus. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that notation, typography, and wording were normalized without changing the mathematical content. These are supplied editorial claims, not independent mathematical verification of this draft.