Projective subscheme | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-06-06

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Projective_variety


00:05:11 1 Variety and scheme structure
00:05:23 1.1 Variety structure
00:20:35 1.2 Projective schemes
00:34:10 2 Relation to complete varieties
00:37:13 3 Examples and basic invariants
00:40:19 3.1 Homogeneous coordinate ring and Hilbert polynomial
00:44:09 3.2 Degree
00:49:37 3.3 The ring of sections
00:52:53 3.4 Projective curves
00:55:40 3.5 Projective hypersurfaces
00:56:14 3.6 Abelian varieties
00:59:26 4 Projections
01:00:07 5 Duality and linear system
01:07:09 6 Cohomology of coherent sheaves
01:15:35 7 Smooth projective varieties
01:24:33 7.1 Serre duality
01:25:08 7.2 Riemann-Roch theorem
01:26:40 8 Hilbert schemes
01:30:42 9 Complex projective varieties
01:34:49 9.1 Relation to complex Kähler manifolds
01:36:15 9.2 GAGA and Chow's theorem
01:37:34 9.3 Complex tori vs. complex abelian varieties
01:40:25 9.4 Kodaira vanishing
01:43:00 9.5 Further topics
01:43:51 10 Related notions
01:44:06 11 See also
01:44:28 12 Notes



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SUMMARY
=======
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space 
  
    
      
        
          
            P
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {P} ^{n}}
   over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of 
  
    
      
        
          
            P
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {P} ^{n}}
  .
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial.
If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

  
    
      
        k
        [
        
          x
          
            0
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        ]
        
          /
        
        I
      
    
    {\displaystyle k[x_{0},\ldots ,x_{n}]/I}
  is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.
Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on X.
A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality. It also leads to the Riemann-Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves is particularly rich, including a classification by the genus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties. Hilbert schemes parametrize closed subschemes of 
  
    
      
        
          
            P
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {P} ^{n}}
   with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also pr ...