14Exx Birational geometry
This subtopic studies birational geometry, where maps that are isomorphisms away from lower-dimensional subsets are used to compare varieties.
Specific topics
14E05 Rational and birational maps
Overview
14E05 studies rational and birational maps in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Rational and birational maps (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for rational and birational maps
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E07 Birational automorphisms, Cremona group
Overview
14E07 studies birational automorphisms, cremona group in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Birational automorphisms, Cremona group (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for birational automorphisms, cremona group
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E08 Rationality questions in algebraic geometry
Overview
14E08 studies rationality questions in algebraic geometry in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Rationality questions in algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for rationality questions in algebraic geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E15 Global theory and resolution of singularities
Overview
14E15 studies global theory and resolution of singularities in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Global theory and resolution of singularities (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for global theory and resolution of singularities
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E16 McKay correspondence
Overview
14E16 studies mckay correspondence in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
McKay correspondence (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for mckay correspondence
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E18 Arcs and motivic integration
Overview
14E18 studies arcs and motivic integration in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Arcs and motivic integration (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for arcs and motivic integration
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E20 Coverings in algebraic geometry
Overview
14E20 studies coverings in algebraic geometry in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Coverings in algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for coverings in algebraic geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E22 Ramification problems in algebraic geometry
Overview
14E22 studies ramification problems in algebraic geometry in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Ramification problems in algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for ramification problems in algebraic geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E25 Embeddings in algebraic geometry
Overview
14E25 studies embeddings in algebraic geometry in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Embeddings in algebraic geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for embeddings in algebraic geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks
14E30 Minimal model program (Mori theory, extremal rays)
Overview
14E30 studies minimal model program (mori theory, extremal rays) in birational geometry. It focuses on birational equivalence, singularity management, and global classification methods that compare algebraic varieties up to rational transformation.
Related Wikipedia Page
Minimal model program (Mori theory, extremal rays) (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for minimal model program (mori theory, extremal rays)
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Classification of higher-dimensional varieties
- Resolution and transformation of singular spaces
- Connections to mirror symmetry, moduli, and arithmetic questions
References
Recommended Textbooks