Wednesday 24 July 2013

Distance Theory Algebraically Supplemented Note 4 Algebraic Cycle

Note 4
Algebraic Cycle


Field     k
Polynomial    x ) = x ]
Algebraic closure k     Polynomial that has one root at least.    
n-dimensional affine space An = kn
n-dimensional complex affine space     Anc = { ( z1, …, zn ) | zi  C }
Finite polynomials over coordinates z1, …, zn     f1 ( z1, …, zn ) , …, fz1, …, zn )
Affine algebraic manifold     f1, …, fm ) = { ( z1, …, zn  Af1 ( z1, …, zn ) = …= fz1, …, zn ) = 0 }
n-dimensional projective space Pn     Continued ratio ( Z1 : … : Zn ) (( Z1 : … : Zn ( 0, …, 0 )
Homogeneous polynomial    F1 Z0 : … : Zn ), …, Fm Z0 : … : Zm )
Projective manifold     F1, …, Fm ) = { ( Z0, …, Zn  PF0 ( Z0, …, Zn ) = …= FmnZ0, …, Zn ) = 0 }
Nonsingular connected projective manifold X     F0, …, Fm )
Homogeneous polynomial    G1 Z0,… : Zn ), …, Gk Z0, … : Z n)
W = F1, …, FmG1, …, Gk )
Algebraic cycle  ∑iaiWi     (Wi is irreducible submanifoldaiZ )

[References]
<Projective space>



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