{ "model": { "rank": 12874, "code": "thing-q648995", "model_id": "vr.tr.algebraic-variety", "name": "algebraic variety", "purpose": "Let an agent explain algebraic varieties and their properties, support learning and problem solving in algebraic geometry, and relate varieties to applications.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q648995/", "tier": 2, "score": 38, "payload": { "layer": "wikidata", "aliases": [ "projective quadric", "affine quadric", "quadric", "algebraic surface", "toric variety", "complex algebraic variety", "algebraic group", "spherical variety", "Severi–Brauer variety", "affine variety", "projective variety", "rational variety", "3-fold", "algebraic manifold", "arithmetic variety", "Chow variety", "complete intersection", "complete variety", "Determinantal variety", "Fake projective space", "Hermitian variety", "Line complex", "Nakajima quiver variety", "Mordellic variety", "polar hypersurface", "Pseudo-canonical variety", "quasiprojective variety", "Rational normal scroll", "ruled variety", "Schubert variety", "secant variety", "Shimura variety", "singular variety", "irreducible variety", "Siegel modular variety", "generalized flag variety", "Klein quartic", "quadric surface", "quadratic curve", "degenerate quadric" ], "aliasCount": 112, "merged": 112, "knownIn": 38, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q648995", "url": "https://www.wikidata.org/wiki/Q648995", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T19:59:16Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q648995", "registryId": "vr.tr.algebraic-variety", "name": "algebraic variety", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q648995/", "model": { "registry_id": "vr.tr.algebraic-variety", "name": "algebraic variety", "purpose": "Let an agent explain algebraic varieties and their properties, support learning and problem solving in algebraic geometry, and relate varieties to applications.", "definition": "The set of solutions of a system of polynomial equations over a field, the central object of algebraic geometry, including affine and projective varieties, curves and surfaces, quadrics and toric varieties; varieties are studied through their coordinate rings, dimension, singularities and maps, and generalised by schemes.", "what_it_is_for": "Algebraic geometry and its applications.", "affordances": [ "explain definitions and examples", "compute dimension, singularities and invariants", "relate varieties to schemes and manifolds", "guide study of algebraic geometry" ], "distinguishing_features": [ "Polynomial equations", "Algebraic and geometric structure", "Dimension and singularities", "Generalised by schemes" ], "appearance": "Not physical; geometric objects defined by equations.", "visual_identification": [ "Zero sets of polynomials", "Affine or projective", "Manifolds are defined by smooth charts, not equations" ], "physical_properties": [], "families_and_kinds": [ "affine varieties", "projective varieties", "curves and surfaces", "quadrics and toric varieties", "complex and arithmetic varieties" ], "related_models": [ { "relation": "is a kind of", "target": "scheme", "why": "category" }, { "relation": "is studied by", "target": "algebraic geometry", "why": "field" }, { "relation": "is related to", "target": "mathematical analysis", "why": "complex analytic methods" }, { "relation": "is related to", "target": "matrix", "why": "linear algebra tools" } ], "identifiers": [], "standards_and_regulation": [ "No regulation; notation conventions in the literature" ], "failure_modes_and_hazards": [ "Confusing varieties with manifolds", "Ignoring the base field", "Misapplying results to singular cases" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "What a variety is.", "rationale": "Equations and fields.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definitions.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definitions.", "questions": [ { "text": "What is an algebraic variety, and how do affine and projective varieties differ?", "kind": "definition" }, { "text": "Which base field is assumed?", "kind": "boundary" } ] } ] }, { "id": "examples", "name": "Examples", "description": "Key examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "What are standard examples, such as conics, quadrics, elliptic curves and toric varieties?", "kind": "provenance" }, { "text": "How is this variety described by equations?", "kind": "definition" } ] } ] } ] }, { "id": "properties", "name": "Properties", "description": "Invariants.", "rationale": "Structure.", "layers": [ { "id": "dimension", "name": "Dimension", "description": "Dimension and degree.", "findings": [ { "id": "dimension-finding", "name": "Dimension", "description": "Dimension.", "questions": [ { "text": "What are the dimension and degree of this variety?", "kind": "measurement" }, { "text": "Is it irreducible?", "kind": "boundary" } ] } ] }, { "id": "singularities", "name": "Singularities", "description": "Singular points.", "findings": [ { "id": "singularities-finding", "name": "Singularities", "description": "Singularities.", "questions": [ { "text": "Where is this variety singular, and how are singularities classified?", "kind": "definition" }, { "text": "How can they be resolved?", "kind": "provenance" } ] } ] } ] }, { "id": "tools", "name": "Tools", "description": "Methods.", "rationale": "Algebra and computation.", "layers": [ { "id": "algebra", "name": "Algebra", "description": "Coordinate rings.", "findings": [ { "id": "algebra-finding", "name": "Algebra", "description": "Coordinate rings.", "questions": [ { "text": "How do coordinate rings and ideals describe the variety?", "kind": "definition" }, { "text": "What does the Nullstellensatz say?", "kind": "definition" } ] } ] }, { "id": "compute", "name": "Compute", "description": "Computational algebra.", "findings": [ { "id": "compute-finding", "name": "Compute", "description": "Computation.", "questions": [ { "text": "Which software and Groebner basis methods compute with this variety?", "kind": "provenance" }, { "text": "How is a computation set up?", "kind": "action" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Study and applications.", "rationale": "Resources.", "layers": [ { "id": "path", "name": "Path", "description": "Learning path.", "findings": [ { "id": "path-finding", "name": "Path", "description": "Learning path.", "questions": [ { "text": "Which texts and prerequisites suit learning algebraic geometry?", "kind": "action" }, { "text": "How do schemes generalise varieties?", "kind": "definition" } ] } ] }, { "id": "applications", "name": "Applications", "description": "Uses.", "findings": [ { "id": "applications-finding", "name": "Applications", "description": "Applications.", "questions": [ { "text": "How are varieties used in cryptography, coding theory, physics and statistics?", "kind": "provenance" }, { "text": "Which examples are notable?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should schemes be a separate entry?", "How should textbooks be linked?", "How should computational tools be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a algebraic variety, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain algebraic varieties and their properties, support learning and problem solving in algebraic geometry, and relate varieties to applications.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: projective quadric, affine quadric, quadric, algebraic surface, toric variety, complex algebraic variety, algebraic group, spherical variety, Severi–Brauer variety, affine variety, projective variety, rational variety.", "112 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 38 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a algebraic variety, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "projective quadric", "affine quadric", "quadric", "algebraic surface", "toric variety", "complex algebraic variety", "algebraic group", "spherical variety", "Severi–Brauer variety", "affine variety", "projective variety", "rational variety" ] } ], "standing": "Described in 38 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 112 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one algebraic variety from another, and a algebraic variety from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a algebraic variety sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a algebraic variety from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a algebraic variety passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a algebraic variety between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a algebraic variety, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a algebraic variety is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What algebraic variety is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of algebraic variety are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is algebraic variety operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of algebraic variety and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of algebraic variety?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as algebraic variety?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a algebraic variety came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a algebraic variety and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a algebraic variety is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a algebraic variety, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a algebraic variety, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a algebraic variety is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }