Algebraic Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of algebraic geometry within the Kireth Research Semantic Library. Algebraic geometry studies geometric objects defined by polynomial equations and the algebraic structures attached to them. It connects geometry with commutative algebra, topology, number theory and complex analysis through affine varieties, projective varieties, schemes, morphisms, coordinate rings, ideals, Zariski topology, dimension, singularities, divisors, line bundles and cohomological structure. This ontology supports classical and modern algebraic-geometric reasoning while preserving the distinction between affine, projective and scheme-theoretic viewpoints.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Algebraic Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Advanced Geometry / Algebraic Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/advanced-geometry/algebraic-geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = AlgebraicGeometry
 SYMBOLIC_FORMS = [
  "V(I)",
  "I(X)",
  "Spec(R)",
  "Proj(S)"
 ]
 ENTITY_CLASS = PolynomialGeometricStructureFramework
 PRIMARY_CONTEXT = AdvancedGeometry
 DEFINITION = "Algebraic geometry is the study of geometric spaces defined by polynomial equations and the algebraic structures associated with them."
 SEMANTIC_ROLE = "Provides the semantic framework connecting polynomial systems, ideals, coordinate rings, varieties, schemes, morphisms, dimension, singularities and projective structure."
}

SCOPE {
 INCLUDE = [
  AlgebraicGeometry,
  AffineVariety,
  ProjectiveVariety,
  AlgebraicSet,
  IrreducibleVariety,
  CoordinateRing,
  PolynomialRing,
  Ideal,
  PrimeIdeal,
  MaximalIdeal,
  RadicalIdeal,
  VanishingIdeal,
  ZariskiTopology,
  RegularFunction,
  RationalFunction,
  Morphism,
  RationalMap,
  Isomorphism,
  BirationalEquivalence,
  Scheme,
  AffineScheme,
  Spectrum,
  StructureSheaf,
  Sheaf,
  LocalRing,
  Dimension,
  KrullDimension,
  SingularPoint,
  SmoothPoint,
  TangentSpace,
  Divisor,
  LineBundle,
  ProjectiveSpace,
  HomogeneousPolynomial,
  HomogeneousIdeal,
  ProjConstruction,
  HilbertPolynomial,
  Degree,
  IntersectionMultiplicity
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralPolynomialAlgebraWithoutGeometricInterpretation,
  DifferentialGeometryWithoutPolynomialStructure,
  PureCommutativeAlgebraWithoutGeometricObject,
  ArithmeticGeometryUnlessExplicit
 ]

 DISAMBIGUATION_RULE =
  "When geometric objects are defined by polynomial equations, ideals, coordinate rings, spectra, projective homogeneous coordinates or scheme-theoretic structure, interpret the context as algebraic geometry."
}

CORE_CONCEPTS {

 AlgebraicGeometry {
  definition = "The geometry of polynomially defined spaces and their algebraic invariants."
 }

 AffineVariety {
  definition = "An irreducible algebraic subset of affine space over a specified field."
 }

 ProjectiveVariety {
  definition = "An irreducible algebraic subset of projective space defined by homogeneous polynomial equations."
 }

 AlgebraicSet {
  definition = "A common zero locus of a family of polynomials."
 }

 CoordinateRing {
  definition = "The quotient k[x_1,...,x_n]/I(X) associated with an affine algebraic set X."
 }

 Ideal {
  definition = "A subset of a ring closed under addition and multiplication by arbitrary ring elements."
 }

 VanishingIdeal {
  symbol = "I(X)"
  definition = "The ideal of all polynomials vanishing on a set X."
 }

 ZariskiTopology {
  definition = "The topology whose closed sets are algebraic sets."
 }

 Morphism {
  definition = "A structure-preserving map locally induced by polynomial or regular functions."
 }

 Scheme {
  definition = "A locally ringed space locally isomorphic to the spectrum of a commutative ring."
 }

 Spectrum {
  symbol = "Spec(R)"
  definition = "The set of prime ideals of a ring R equipped with the Zariski topology and structure sheaf."
 }

 StructureSheaf {
  definition = "A sheaf assigning local rings of functions to open subsets of a scheme."
 }

 SingularPoint {
  definition = "A point where the local geometric structure fails the relevant smoothness criterion."
 }

 TangentSpace {
  definition = "A linear approximation derived from local equations, derivations or the maximal ideal modulo its square."
 }

 Divisor {
  definition = "A formal codimension-one geometric object, commonly expressed as a combination of irreducible subvarieties."
 }
}

RELATIONS {
 AlgebraicGeometry --is_branch_of--> AdvancedGeometry
 AlgebraicSet --is_zero_locus_of--> PolynomialFamily
 VanishingIdeal --is_ideal_of--> AlgebraicSet
 CoordinateRing --is_quotient_of--> PolynomialRing
 AffineVariety --has--> CoordinateRing
 ProjectiveVariety --is_subspace_of--> ProjectiveSpace
 ProjectiveVariety --is_defined_by--> HomogeneousIdeal
 ZariskiTopology --has_closed_sets--> AlgebraicSets
 Morphism --maps_between--> AlgebraicVarieties
 Isomorphism --is_special_case_of--> Morphism
 BirationalEquivalence --relates--> Varieties
 Scheme --is_locally_modelled_on--> AffineScheme
 AffineScheme --is--> Spectrum
 Spectrum --consists_of--> PrimeIdeal
 StructureSheaf --assigns--> LocalRing
 SingularPoint --contrasts_with--> SmoothPoint
 TangentSpace --is_attached_to--> Point
 Divisor --is_codimension_one_structure_on--> Variety
}

OPERATORS {
 ZERO_LOCUS(I,k^n) :=
  {x in k^n : f(x)=0 for_all_f_in_I}

 VANISHING_IDEAL(X) :=
  {f in k[x_1,...,x_n] : f(x)=0 for_all_x_in_X}

 RADICAL(I) :=
  {f : exists_n>=1 with f^n in I}

 COORDINATE_RING(X) :=
  k[x_1,...,x_n]/I(X)

 SPEC(R) :=
  set_of_prime_ideals_of_R

 IS_IRREDUCIBLE(X) :=
  X_cannot_be_written_as_union_of_two_proper_closed_subsets

 IS_MORPHISM(f,X,Y) :=
  f_is_regular_locally_and_respects_structure

 ZARISKI_CLOSURE(S) :=
  ZERO_LOCUS(VANISHING_IDEAL(S))

 TANGENT_SPACE_JACOBIAN(X,p) :=
  kernel_of_Jacobian_matrix_of_defining_equations_at_p
  under_affine_variety_context

 IS_SMOOTH_POINT(X,p) :=
  tangent_dimension_equals_local_dimension

 PROJECTIVISE_HOMOGENEOUS_IDEAL(I) :=
  projective_zero_locus_of_I

 PROJ(S) :=
  homogeneous_prime_ideals_not_containing_irrelevant_ideal
  with_structure_sheaf
}

DEFINITIONS {
 D1 = "Algebraic sets are zero loci of polynomial families."
 D2 = "The Zariski topology takes algebraic sets as closed sets."
 D3 = "Coordinate rings encode affine algebraic sets algebraically."
 D4 = "Irreducible affine varieties correspond to prime ideals under standard hypotheses."
 D5 = "Projective varieties are defined by homogeneous equations."
 D6 = "Schemes generalise varieties by retaining nilpotent and arithmetic structure."
 D7 = "Morphisms are the structure-preserving maps of algebraic geometry."
 D8 = "Smoothness can be tested by Jacobian rank in suitable finite-type settings."
 D9 = "Birationally equivalent varieties share isomorphic dense open subsets."
 D10 = "Dimension can be described algebraically by chains of prime ideals."
}

AXIOMS {
 A1_HILBERT_NULLSTELLENSATZ_CONTEXT:
  Over an algebraically closed field, radical ideals and affine algebraic sets correspond contravariantly

 A2_ZARISKI:
  Algebraic sets define the closed sets of the Zariski topology

 A3_COORDINATE_DUALITY:
  Affine geometry and finitely generated reduced coordinate algebras are contravariantly related under standard hypotheses

 A4_PROJECTIVE_HOMOGENEITY:
  Projective algebraic sets are defined by homogeneous ideals

 A5_SCHEME_LOCALITY:
  Schemes are locally affine

 A6_MORPHISM_LOCALITY:
  Regularity of scheme morphisms is local on source and target

 A7_DIMENSION:
  Krull dimension is the supremum of lengths of chains of prime ideals

 A8_TANGENT_SMOOTHNESS:
  Tangent-space dimension detects singularity under suitable finite-type hypotheses
}

REASONING {

 R1_ZERO_LOCUS {
  INPUT = polynomial_family_or_ideal
  SOLVE = common_zero_conditions
  RETURN = algebraic_set
 }

 R2_VANISHING_IDEAL {
  INPUT = subset_X
  COLLECT = polynomials_vanishing_on_X
  RETURN = I(X)
 }

 R3_AFFINE_VARIETY_ANALYSIS {
  INPUT = affine_algebraic_set_X
  COMPUTE = vanishing_ideal
  TEST = irreducibility
  COMPUTE = coordinate_ring
  RETURN = variety_structure
 }

 R4_PROJECTIVE_VARIETY_ANALYSIS {
  INPUT = homogeneous_equations
  VERIFY = homogeneity
  FORM = projective_zero_locus
  RETURN = projective_variety
 }

 R5_MORPHISM_TEST {
  INPUT = candidate_map_f
  VERIFY = coordinate_functions_regular_on_local_charts
  RETURN = MorphismOrNot
 }

 R6_SINGULARITY_TEST {
  INPUT = defining_equations_and_point_p
  COMPUTE = Jacobian_matrix_at_p
  COMPUTE = tangent_space_dimension
  COMPARE = expected_local_dimension
  RETURN = SmoothOrSingular
 }

 R7_SCHEME_LOCALISATION {
  INPUT = ring_R_and_prime_p
  LOCALISE = R_at_p
  RETURN = local_ring_at_p
 }

 R8_DIMENSION {
  INPUT = ring_or_variety
  IF ring_context
   COMPUTE = maximal_prime_chain_length
  ELSE
   USE = coordinate_ring_dimension
  RETURN = dimension
 }

 R9_BIRATIONAL_REASONING {
  INPUT = varieties_X_Y
  SEARCH = dense_open_subsets_U_V
  TEST = isomorphism_U_to_V
  RETURN = BirationallyEquivalentOrNot
 }

 R10_DIVISOR_REASONING {
  INPUT = codimension_one_data
  CLASSIFY = Weil_or_Cartier_context
  VERIFY = local_principal_or_cycle conditions
  RETURN = divisor_structure
 }
}

CONSTRAINTS {
 C1 = "Base field or base scheme should be identified."
 C2 = "Classical variety definitions may depend on algebraic closedness."
 C3 = "Algebraic set and irreducible variety must not be conflated."
 C4 = "Radical, prime and maximal ideals have distinct geometric meanings."
 C5 = "Projective equations must be homogeneous."
 C6 = "Scheme structure retains nilpotent information that underlying point sets lose."
 C7 = "Smoothness criteria depend on characteristic and finite-type assumptions."
 C8 = "Birational equivalence is weaker than isomorphism."
 C9 = "Zariski topology is generally much coarser than Euclidean topology."
 C10 = "Dimension conventions must distinguish algebraic, Krull and ambient dimension."
}

SEMANTIC_EQUIVALENCE {
 "AlgebraicGeometry" <=> "GeometryOfPolynomiallyDefinedSpaces"
 "AffineAlgebraicSet" <=> "PolynomialZeroLocus"
 "CoordinateRing" <=> "PolynomialFunctionsModuloVanishingIdeal"
 "Spectrum" <=> "PrimeIdealSpace"
 "ProjectiveVariety" <=> "HomogeneousPolynomialZeroLocus"
 "BirationalEquivalence" <=> "IsomorphismOnDenseOpenSubsets"
}

VALIDATION {
 V1 = "Base field or ring is identified."
 V2 = "Projective equations are homogeneous."
 V3 = "Coordinate ring uses the correct vanishing ideal."
 V4 = "Irreducibility claims correspond to prime-ideal conditions where applicable."
 V5 = "Scheme morphism reasoning is local."
 V6 = "Jacobian smoothness test uses correct hypotheses."
 V7 = "Birational equivalence is not confused with global isomorphism."
 V8 = "Nilpotent scheme structure is not discarded when relevant."
 V9 = "Dimension interpretation is explicitly stated."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ALGEBRAIC_GEOMETRY_DEFINITION = 0.99
 ZERO_LOCUS_REASONING = 0.99
 IDEAL_VARIETY_REASONING = 0.98
 PROJECTIVE_REASONING = 0.99
 SCHEME_REASONING = 0.97
 SINGULARITY_REASONING = 0.98
 DIMENSION_REASONING = 0.98
 BIRATIONAL_REASONING = 0.97

 POLICY =
  "Reduce confidence when base field, characteristic, reducedness, irreducibility, finite-type hypotheses, scheme-versus-variety context or dimension convention are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  AlgebraicGeometryReasoning,
  PolynomialZeroLocusReasoning,
  IdealVarietyReasoning,
  CoordinateRingReasoning,
  ZariskiTopologyReasoning,
  MorphismReasoning,
  ProjectiveVarietyReasoning,
  SchemeReasoning,
  DimensionReasoning,
  SingularityReasoning,
  TangentSpaceReasoning,
  BirationalReasoning,
  DivisorReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "algebraic geometry" -> AlgebraicGeometry
  "affine variety" -> AffineVariety
  "projective variety" -> ProjectiveVariety
  "algebraic set" -> AlgebraicSet
  "coordinate ring" -> CoordinateRing
  "vanishing ideal" -> VanishingIdeal
  "Zariski topology" -> ZariskiTopology
  "scheme" -> Scheme
  "spectrum" -> Spectrum
  "singular point" -> SingularPoint
  "birational" -> BirationalEquivalence
 }

 INTERPRETATION_ORDER = [
  detect_algebraic_geometry_context,
  identify base_field_or_ring_and_affine_projective_or_scheme setting,
  identify equations_ideals_coordinate_rings_morphisms_dimension_or_singularity goal,
  classify classical_variety_or_scheme_theoretic pathway,
  apply algebraic_geometric correspondence,
  verify hypotheses_homogeneity_reducedness_irreducibility_and_dimension conventions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("algebraic geometry")
  OR polynomial_zero_locus_geometry_context
  THEN activate(AlgebraicGeometryOntology)

 E2:
  IF affine_zero_locus_requested
  THEN compute V(I)

 E3:
  IF vanishing_ideal_requested
  THEN compute I(X)

 E4:
  IF projective_variety_context
  THEN require homogeneous_equations

 E5:
  IF affine_variety_irreducibility_requested
  THEN relate irreducibility_to_prime ideal under suitable hypotheses

 E6:
  IF scheme_context
  THEN retain structure_sheaf_and_local_ring semantics

 E7:
  IF singularity_requested
  THEN apply Jacobian_or_local_ring criterion appropriate_to_context

 E8:
  IF birational_equivalence_requested
  THEN search for isomorphic_dense_open subsets

 E9:
  IF dimension_requested
  THEN specify Krull_or_geometric dimension convention
}

RECURSION {
 ENTRY = AlgebraicGeometry

 EXPAND(AlgebraicGeometry) -> [
  AffineVariety,
  ProjectiveVariety,
  AlgebraicSet,
  IrreducibleVariety,
  CoordinateRing,
  PolynomialRing,
  Ideal,
  PrimeIdeal,
  RadicalIdeal,
  VanishingIdeal,
  ZariskiTopology,
  Morphism,
  RationalMap,
  BirationalEquivalence,
  Scheme,
  AffineScheme,
  Spectrum,
  StructureSheaf,
  LocalRing,
  Dimension,
  SingularPoint,
  SmoothPoint,
  TangentSpace,
  Divisor,
  LineBundle
 ]

 EXPAND(AffineGeometryStructure) -> [
  ZeroLocus,
  VanishingIdeal,
  CoordinateRing,
  IrreducibleComponent,
  RegularFunction
 ]

 EXPAND(ProjectiveGeometryStructure) -> [
  HomogeneousCoordinates,
  HomogeneousPolynomial,
  HomogeneousIdeal,
  ProjectiveSpace,
  ProjConstruction
 ]

 EXPAND(SchemeStructure) -> [
  Spectrum,
  PrimeIdeal,
  StructureSheaf,
  LocalRing,
  Stalk,
  MorphismOfSchemes
 ]

 STOP_CONDITION =
  "Stop when the polynomial equations, ideals, coordinate ring, variety or scheme structure, morphism, dimension, singularity, divisor or birational relation is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Advanced Geometry"

 IMMEDIATE_CONTEXTS = [
  "Advanced Geometry",
  "Projective Geometry Ontology",
  "Affine Geometry Ontology",
  "Commutative Algebra Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Advanced Geometry",
  "Projective Geometry Ontology",
  "Affine Geometry Ontology",
  "Topology Ontology",
  "Linear Algebra Ontologies",
  "Polynomial Ontology",
  "Ring Theory Ontologies",
  "Ideal Ontologies",
  "Commutative Algebra Ontologies",
  "Number Theory Ontologies",
  "Complex Geometry Ontologies",
  "Differential Geometry Ontologies"
 ]

 ROUTE_RULES = [
  "Algebraic Geometry -> Advanced Geometry for parent structural semantics",
  "Algebraic Geometry -> Projective Geometry for projective varieties and homogeneous coordinates",
  "Algebraic Geometry -> Affine Geometry for affine spaces and coordinate charts",
  "Algebraic Geometry -> Commutative Algebra for ideals, spectra and local rings",
  "Algebraic Geometry -> Polynomial Ontology for defining equations",
  "Algebraic Geometry -> Topology for Zariski spaces",
  "Algebraic Geometry -> Number Theory for arithmetic varieties",
  "Algebraic Geometry -> Complex Geometry for complex algebraic varieties",
  "Algebraic Geometry -> Differential Geometry for smooth-locus comparisons"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible varieties, schemes, invariants or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve polynomial-zero-locus semantics.",
  "Preserve ideal-variety correspondence with correct hypotheses.",
  "Preserve affine-versus-projective distinction.",
  "Preserve homogeneous-equation requirements.",
  "Preserve scheme local-ring and sheaf semantics.",
  "Preserve singular-versus-smooth distinction.",
  "Preserve birational-versus-isomorphic distinction.",
  "Preserve machine-readable algebraic-geometry semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Algebraic geometry studies geometric spaces defined by polynomial equations together with their algebraic structures."

 AFFINE_SUMMARY =
  "Affine algebraic sets are common zero loci of polynomial families."

 COORDINATE_RING_SUMMARY =
  "The affine coordinate ring is k[x_1,...,x_n]/I(X)."

 ZARISKI_SUMMARY =
  "Zariski-closed sets are algebraic sets."

 PROJECTIVE_SUMMARY =
  "Projective varieties are defined by homogeneous polynomial equations in projective space."

 SCHEME_SUMMARY =
  "A scheme is a locally ringed space locally isomorphic to Spec(R)."

 DIMENSION_SUMMARY =
  "Krull dimension is measured by chains of prime ideals."

 SINGULARITY_SUMMARY =
  "Singular points are locations where local algebraic geometry fails the relevant smoothness criterion."

 BIRATIONAL_SUMMARY =
  "Birationally equivalent varieties contain isomorphic dense open subsets."
}

FLOW {
 INPUT
 -> detect_algebraic_geometry_context
 -> identify base_field_or_ring_affine_projective_scheme_and_equation data
 -> identify zero_locus_ideal_coordinate_ring_morphism_dimension_singularity_divisor_or_birational goal
 -> classify classical_variety_or_scheme pathway
 -> apply polynomial_ideal_ring_sheaf_and_local reasoning
 -> verify algebraic_closedness_homogeneity_reducedness_irreducibility_finite_type_and_dimension assumptions
 -> route to advanced_geometry projective_geometry affine_geometry commutative_algebra topology polynomial number_theory complex_or_differential_geometry ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Algebraic Geometry Ontology defines the geometry of polynomially specified spaces and connects affine and projective varieties with ideals, coordinate rings, Zariski topology, morphisms, spectra, schemes, local rings, dimension, tangent spaces, singularities, divisors and birational equivalence. It preserves the distinction between classical variety theory and modern scheme-theoretic structure."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-ALGEBRAIC-GEOMETRY-ONTOLOGY-V1"
}

}

END_FLOW