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ZariskiFrames

(Co)frames/Locales of Zariski closed/open subsets of affine, projective, or toric varieties

2024.12-01

6 December 2024

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Tom Kuhmichel
Email: tom.kuhmichel@student.uni-siegen.de
Homepage: https://github.com/TKuh
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Markus Lange-Hegermann
Email: markus.lange-hegermann@th-owl.de
Homepage: https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/
Address:
Markus Lange-Hegermann
Hochschule Ostwestfalen-Lippe
Liebigstraße 87
32657 Lemgo
Germany

Contents

1 (Co)frames/Locales of Zariski closed/open subsets

1.1-1 InfoZariskiFrames

1.2 GAP Categories

  1.2-1 IsZariskiFrameOrCoframe
  1.2-2 IsObjectInZariskiFrameOrCoframe
  1.2-3 IsMorphismInZariskiFrameOrCoframe

1.3-1 IsOpen
1.3-2 IsClosed

  1.4-1 UnderlyingRing
  1.4-2 BaseOfFibration
  1.4-3 BaseOfFibration
  1.4-4 Closure
  1.4-5 ParametrizedObject
  1.4-6 ParametrizedObject
  1.4-7 ParametrizedObject
  1.4-8 LocallyClosedPart
  1.4-9 DistinguishedLocallyClosedPart
  1.4-10 DistinguishedLocallyClosedPart
  1.4-11 DistinguishedLocallyClosedPart
  1.4-12 AffineApproximation
  1.4-13 CanonicalObject
  1.4-14 RabinowitschCover

  1.5-1 AClosedSuperset
  1.5-2 BestUnderlyingColumn
  1.5-3 NormalizeObject
  1.5-4 StandardizeObject
  1.5-5 IsSubset
  1.5-6 RingEpimorphismOfClosedSuperset
  1.5-7 RingEpimorphismOfClosure
  1.5-8 Pullback
  1.5-9 EmbedInSmallerAmbientSpaceByEmbeddingAClosedSuperset
  1.5-10 EmbedInSmallerAmbientSpace
  1.5-11 AClosedSingleton
  1.5-12 PseudoIteratorOfClosedSingletons
  1.5-13 RingEpimorphismOfAClosedPoint
  1.5-14 AClosedPoint
  1.5-15 PseudoIteratorOfClosedPoints

2 Frames of Zariski open subsets

2.1 GAP Categories

  2.1-1 IsZariskiFrame
  2.1-2 IsObjectInZariskiFrame
  2.1-3 IsMorphismInZariskiFrame

  2.2-1 DimensionOfComplement
  2.2-2 DegreeOfComplement
  2.2-3 ComplementAsClosedSubset
  2.2-4 AsDifferenceOfClosed

3 Coframes of Zariski closed subsets

3.1 GAP Categories

  3.1-1 IsZariskiCoframe
  3.1-2 IsObjectInZariskiCoframe
  3.1-3 IsMorphismInZariskiCoframe

3.2-1 IsIrreducibleObjectInZariskiCoframe

  3.3-1 Dimension
  3.3-2 FactorizeObject
  3.3-3 IrreducibleComponents

3.4-1 \[\]
3.4-2 KnownFactors

4 (Co)frames/Locales of Zariski closed/open subsets in affine varieties

4.1 GAP Categories

4.1-1 IsObjectInZariskiFrameOrCoframeOfAnAffineVariety
4.1-2 IsMorphismInZariskiFrameOrCoframeOfAnAffineVariety

4.2-1 DistinguishedQuasiAffineSet

5 The frame of Zariski open subsets in an affine variety

5.1 GAP Categories

  5.1-1 IsZariskiFrameOfAnAffineVariety
  5.1-2 IsObjectInZariskiFrameOfAnAffineVariety
  5.1-3 IsMorphismInZariskiFrameOfAnAffineVariety

5.2 Constructors

  5.2-1 ZariskiFrameOfAffineSpectrum
  5.2-2 OpenSubsetOfSpec
  5.2-3 OpenSubsetOfSpecByRadicalColumn
  5.2-4 OpenSubsetOfSpecByStandardColumn

6 The coframe of Zariski closed subsets in an affine variety

6.1 GAP Categories

  6.1-1 IsZariskiCoframeOfAnAffineVariety
  6.1-2 IsObjectInZariskiCoframeOfAnAffineVariety
  6.1-3 IsMorphismInZariskiCoframeOfAnAffineVariety

6.2 Constructors

  6.2-1 ZariskiCoframeOfAffineSpectrum
  6.2-2 ClosedSubsetOfSpec
  6.2-3 ClosedSubsetOfSpecByRadicalColumn
  6.2-4 ClosedSubsetOfSpecByStandardColumn
  6.2-5 TangentSpaceAtPoint
  6.2-6 TangentSpaceAtPoint

7 Relative affine varieties

  7.1-1 GraphOfRingMorphism
  7.1-2 FunctorClosureOfProjectionBetweenZariskiCoframes
  7.1-3 ClosureOfProjection
  7.1-4 ClosureOfImage
  7.1-5 FunctorPreimageOfProjectionBetweenZariskiCoframes

  7.2-1 PreimageOfProjection
  7.2-2 PreimageOfProjection
  7.2-3 FiberOfProjectionOverBasePoint
  7.2-4 FiberOfProjectionOverBasePoint
  7.2-5 TangentSpaceOfFiberAtPoint
  7.2-6 TangentSpaceOfFiberAtPoint
  7.2-7 DimensionsOfFibrationAtClosedPoint
  7.2-8 DimensionsOfFibrationAtClosedPoint
  7.2-9 EmbeddedComplementOfTangentSpaceOfFiberAtPoint
  7.2-10 EmbeddedComplementOfTangentSpaceOfFiberAtPoint
  7.2-11 ClosedSubsetWithGenericallyZeroDimensionalFibers
  7.2-12 ClosedSubsetOfBaseWithFreeFibersOverComplementOrEmpty
  7.2-13 ClosedSubsetOfBaseWithFreeFibersOverComplement

8 (Co)frames/Locales of Zariski closed/open subsets in projective varieties

8.1 GAP Categories

  8.1-1 IsObjectInZariskiFrameOrCoframeOfAProjectiveVariety
  8.1-2 IsMorphismInZariskiFrameOrCoframeOfAProjectiveVariety
  8.1-3 UnderlyingListOfSaturatedColumns
  8.1-4 SaturatedUnderlyingColumn

9 The frame of Zariski open subsets in a projective variety

9.1 GAP Categories

  9.1-1 IsZariskiFrameOfAProjectiveVariety
  9.1-2 IsObjectInZariskiFrameOfAProjectiveVariety
  9.1-3 IsMorphismInZariskiFrameOfAProjectiveVariety

9.2 Constructors

  9.2-1 ZariskiFrameOfProj
  9.2-2 OpenSubsetOfProj
  9.2-3 OpenSubsetOfProjByRadicalColumn
  9.2-4 OpenSubsetOfProjByStandardColumn

10 The coframe of Zariski closed subsets in a projective variety

10.1 GAP Categories

  10.1-1 IsZariskiCoframeOfAProjectiveVariety
  10.1-2 IsObjectInZariskiCoframeOfAProjectiveVariety
  10.1-3 IsMorphismInZariskiCoframeOfAProjectiveVariety

10.2 Attributes

10.2-1 UnderlyingClosedSubsetOfSpec

10.3 Constructors

  10.3-1 ZariskiCoframeOfProj
  10.3-2 ClosedSubsetOfProj
  10.3-3 ClosedSubsetOfProjByRadicalColumn
  10.3-4 ClosedSubsetOfProjByStandardColumn

11 Relative projective varieties

11.1 Attributes

  11.1-1 PointsAtInfinityOfFiberwiseProjectiveClosure
  11.1-2 FunctorClosedProjectionBetweenZariskiCoframes
  11.1-3 ImageOfProjection
  11.1-4 ProjectionOfPointsAtInfinityOfFiberwiseProjectiveClosure

12 Constructible image

12.1 Info Class

12.1-1 InfoConstructibleImage

12.2 Operations

  12.2-1 DecreaseCodimensionByFixingVariables
  12.2-2 ClosedSubsetWithGenericallyZeroDimensionalFibersOnComponentOfProjection
  12.2-3 LocallyClosedApproximationOfProjection
  12.2-4 LocallyClosedApproximationOfImage
  12.2-5 LocallyClosedApproximationOfProjectionViaGenericFreeness
  12.2-6 LocallyClosedApproximationOfImageViaGenericFreeness
  12.2-7 ConstructibleProjection
  12.2-8 ConstructibleImage
  12.2-9 Visualize

13 Further examples

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