"Elliptic-Parabolic-Hyperbolic trichotomy in mathematics"의 두 판 사이의 차이

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==algebraic geometry==
 
==algebraic geometry==
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* Let $X$ be a smooth complex projective variety. There are three main types of varieties.  
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* 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 $−K_X$ is ample. (We recall the definition of ampleness in section 2.)
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# Fano. This means that <math>−K_X</math> is ample. (We recall the definition of ampleness in section 2.)
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# Calabi-Yau. We define this to mean that $K_X$ is numerically trivial.
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# Calabi-Yau. We define this to mean that <math>K_X</math> is numerically trivial.
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# ample canonical bundle. This means that $K_X$ is ample; it implies that $X$ is of general type.”
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# ample canonical bundle. This means that <math>K_X</math> is ample; it implies that <math>X</math> is of general type.”
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* Here, for $X$ of complex dimension $n$, the canonical bundle $K_X$ is the line bundle $\Omega^n_X$ of $n$-forms.  
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* 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 $−K_X$ for the dual line bundle $K^∗_X$, the determinant of the tangent bundle.
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* 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==
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* 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]]
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[[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
  1. Fano. This means that \(−K_X\) is ample. (We recall the definition of ampleness in section 2.)
  2. Calabi-Yau. We define this to mean that \(K_X\) is numerically trivial.
  3. 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.
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