Very Ample Internal Strongly Minimal Sets

Definitions and mathematical statement

A definable set $S$ is strongly minimal if it is infinite and, in every elementary extension, every definable subset of $S$, allowing parameters, is finite or cofinite. Definability is understood relative to an ambient structure and a choice of allowed parameters; these are not specified in the supplied extract.

Question as supplied. For the object $M$ intended in the source, does there exist a strongly minimal set $S$ that is internal to $M$ and very ample? More specifically, does there exist a definable equivalence relation \[ E\subseteq M\times M \] such that the quotient \[ M/E=\{[a]_E:a\in M\},\qquad [a]_E=\{b\in M:E(a,b)\}, \] is very ample? Here an equivalence relation is reflexive, symmetric, and transitive; the quotient is regarded as an imaginary sort in the ambient model-theoretic setting.

Unresolved scope. The supplied record does not identify $M$, specify its hypotheses, or state whether the question concerns a particular $M$ or every $M$ in some class. It also does not define the source's technical notion of very ampleness or specify the parameter conventions for internality and definability. Those terms are therefore retained without substituting an unverified definition. The phrase "more specifically" is preserved as the source's relationship between the two questions; no equivalence between them is asserted. These omissions prevent a fully specified standalone conjecture.

Reported background, separate from the question. The original extract also reports the following theorem: let $K$ be an algebraically closed field and let $(G,\cdot)$ be a divisible one-dimensional algebraic group over $K$. Let $G_{\mathrm{lin}}$ be the expansion of $(G,\cdot)$ by all $K$-definable endomorphisms of $G$. Then every $K$-relic properly expanding $G_{\mathrm{lin}}$ is full. The extract presents this as answering "Martin's question," but does not identify Martin or define "$K$-relic," "properly expanding," or "full." These hypotheses are not silently imposed on $M$.

The extract further expresses the suspicion that Hrushovski's amalgamation construction technique may provide a counterexample, at least to the "weaker form" of the question. This is a reported suspicion, not an established counterexample; the supplied text does not unambiguously identify that weaker form.

Short English statement

The source asks whether its unspecified object $M$ internally supports a very ample strongly minimal set, and more specifically whether a definable quotient $M/E$ can be very ample. Missing hypotheses on $M$ and missing technical conventions prevent a complete standalone formulation. The accompanying classification theorem and suggested amalgamation counterexample are retained as reported context. The supplied open label requires independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005357$, title Very Ample Internal Strongly Minimal Sets, exact source problem code catalogue entry. The record assigns the problem to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2023; no proposer is supplied.
• Supplied report citation: Model Theory: Combinatorics, Groups, Valued Fields and Neostability (2023), Oberwolfach Reports, pp. 91--149, DOI 10.4171/owr/2023/2. https://doi.org/10.4171/owr/2023/2. The record describes this report as documenting workshop discussions and identifies it as the source of the open question.
• Supplied literature source: Castle--Hasson--Ye, related classification results. https://doi.org/10.2140/mt.2024.3.213. No further bibliographic details or precise applicable results are supplied.
• Supplied literature assessment, checked on 2026-08-22: the record labels the question open, says that the report leaves the internal very-ampleness question open and proves only special cases, and reports that no later primary source resolving the general question was located by that date. The listed sources checked are the Oberwolfach report and the Castle--Hasson--Ye reference above. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.