"Elliptic-Parabolic-Hyperbolic trichotomy in mathematics"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 6개는 보이지 않습니다) | |||
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* spherical-Euclidean-hyperbolic | * spherical-Euclidean-hyperbolic | ||
* finite-affine-indefinite | * finite-affine-indefinite | ||
| + | * {{수학노트|url=리만_사상_정리_Riemann_mapping_theorem_and_the_uniformization_theorem}} | ||
| + | * {{수학노트|url=이차곡선(원뿔곡선)}} | ||
==algebraic geometry== | ==algebraic geometry== | ||
| − | * Let | + | * Let <math>X</math> be a smooth complex projective variety. There are three main types of varieties. |
* Not every variety is of one of these three types, but minimal model theory relates every variety to one of these extreme types | * Not every variety is of one of these three types, but minimal model theory relates every variety to one of these extreme types | ||
| − | # Fano. This means that | + | # Fano. This means that <math>−K_X</math> is ample. (We recall the definition of ampleness in section 2.) |
| − | # Calabi-Yau. We define this to mean that | + | # Calabi-Yau. We define this to mean that <math>K_X</math> is numerically trivial. |
| − | # ample canonical bundle. This means that | + | # ample canonical bundle. This means that <math>K_X</math> is ample; it implies that <math>X</math> is of general type.” |
| − | * Here, for | + | * Here, for <math>X</math> of complex dimension <math>n</math>, the canonical bundle <math>K_X</math> is the line bundle <math>\Omega^n_X</math> of <math>n</math>-forms. |
| − | * We write | + | * We write <math>−K_X</math> for the dual line bundle <math>K^∗_X</math>, the determinant of the tangent bundle. |
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==articles== | ==articles== | ||
| + | * Terragni, T. “Data about Hyperbolic Coxeter Systems.” arXiv:1503.08764 [math], March 30, 2015. http://arxiv.org/abs/1503.08764. | ||
* Rastegar, Arash. ‘EPH-Classifications in Geometry, Algebra, Analysis and Arithmetic’. arXiv:1503.07859 [math], 26 March 2015. http://arxiv.org/abs/1503.07859. | * Rastegar, Arash. ‘EPH-Classifications in Geometry, Algebra, Analysis and Arithmetic’. arXiv:1503.07859 [math], 26 March 2015. http://arxiv.org/abs/1503.07859. | ||
* Totaro, Burt. ‘Algebraic Surfaces and Hyperbolic Geometry’. ArXiv E-Prints 1008 (1 August 2010): 3825. | * Totaro, Burt. ‘Algebraic Surfaces and Hyperbolic Geometry’. ArXiv E-Prints 1008 (1 August 2010): 3825. | ||
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[[분류:abstract concepts]] | [[분류:abstract concepts]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 16일 (월) 10:05 기준 최신판
introduction
algebraic geometry
- Let \(X\) be a smooth complex projective variety. There are three main types of varieties.
- Not every variety is of one of these three types, but minimal model theory relates every variety to one of these extreme types
- Fano. This means that \(−K_X\) is ample. (We recall the definition of ampleness in section 2.)
- Calabi-Yau. We define this to mean that \(K_X\) is numerically trivial.
- ample canonical bundle. This means that \(K_X\) is ample; it implies that \(X\) is of general type.”
- Here, for \(X\) of complex dimension \(n\), the canonical bundle \(K_X\) is the line bundle \(\Omega^n_X\) of \(n\)-forms.
- We write \(−K_X\) for the dual line bundle \(K^∗_X\), the determinant of the tangent bundle.
memo
articles
- Terragni, T. “Data about Hyperbolic Coxeter Systems.” arXiv:1503.08764 [math], March 30, 2015. http://arxiv.org/abs/1503.08764.
- Rastegar, Arash. ‘EPH-Classifications in Geometry, Algebra, Analysis and Arithmetic’. arXiv:1503.07859 [math], 26 March 2015. http://arxiv.org/abs/1503.07859.
- Totaro, Burt. ‘Algebraic Surfaces and Hyperbolic Geometry’. ArXiv E-Prints 1008 (1 August 2010): 3825.