14Hxx Curves
This subtopic studies curves, covering their geometry, moduli, singularities, maps, and the rich interaction between curve theory and arithmetic.
Specific topics
14H05 Algebraic functions and function fields
Overview
14H05 studies algebraic functions and function fields in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Algebraic functions and function fields (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for algebraic functions and function fields
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H10 Families, moduli of curves
Overview
14H10 studies families, moduli of curves in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Families, moduli of curves (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for families, moduli of curves
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H15 Families, moduli of curves, analytic theory
Overview
14H15 studies families, moduli of curves, analytic theory in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Families, moduli of curves, analytic theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for families, moduli of curves, analytic theory
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H20 Singularities of curves, local rings
Overview
14H20 studies singularities of curves, local rings in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Singularities of curves, local rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for singularities of curves, local rings
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H25 Arithmetic ground fields for curves
Overview
14H25 studies arithmetic ground fields for curves in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Arithmetic ground fields for curves (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for arithmetic ground fields for curves
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H30 Coverings of curves, fundamental group
Overview
14H30 studies coverings of curves, fundamental group in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Coverings of curves, fundamental group (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for coverings of curves, fundamental 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H37 Automorphisms of curves
Overview
14H37 studies automorphisms of curves in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Automorphisms of curves (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for automorphisms of curves
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H40 Jacobians, Prym varieties
Overview
14H40 studies jacobians, prym varieties in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Jacobians, Prym varieties (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for jacobians, prym varieties
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H42 Theta functions and curves; Schottky problem
Overview
14H42 studies theta functions and curves; schottky problem in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Theta functions and curves; Schottky problem (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for theta functions and curves; schottky problem
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H45 Special algebraic curves and curves of low genus
Overview
14H45 studies special algebraic curves and curves of low genus in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Special algebraic curves and curves of low genus (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for special algebraic curves and curves of low genus
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H50 Plane and space curves
Overview
14H50 studies plane and space curves in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Plane and space curves (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for plane and space curves
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H51 Special divisors on curves
Overview
14H51 studies special divisors on curves in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Special divisors on curves (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for special divisors on curves
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H52 Elliptic curves
Overview
14H52 studies elliptic curves in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Elliptic curves (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for elliptic curves
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H55 Riemann surfaces
Overview
14H55 studies riemann surfaces in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Riemann surfaces (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for riemann surfaces
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H57 Dessins d'enfants theory
Overview
14H57 studies dessins d'enfants theory in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Dessins d'enfants theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for dessins d'enfants theory
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H60 Vector bundles on curves and their moduli
Overview
14H60 studies vector bundles on curves and their moduli in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Vector bundles on curves and their moduli (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for vector bundles on curves and their moduli
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H70 Relationships between curves and integrable systems
Overview
14H70 studies relationships between curves and integrable systems in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Relationships between curves and integrable systems (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for relationships between curves and integrable systems
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks
14H81 Relationships between curves and physics
Overview
14H81 studies relationships between curves and physics in algebraic curves. It emphasizes the geometry, arithmetic, and moduli of curves together with the special structures that make one-dimensional varieties central across algebraic geometry.
Related Wikipedia Page
Relationships between curves and physics (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for relationships between curves and physics
- 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 and moduli of curves
- Arithmetic and analytic study of one-dimensional varieties
- Connections to integrable systems, topology, and physics
References
Recommended Textbooks