Distance Theory Algebraically Supplemented
Note 4
Algebraic Cycle
TANAKA Akio
Field k
Polynomial f ( x ) = k [ 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 ) , …, fn ( z1, …, zn )
Affine algebraic manifold V ( f1, …, fm ) = { ( z1, …, zn ) ∈ An | f1 ( z1, …, zn ) = …= fm ( z1, …, 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 V ( F1, …, Fm ) = { ( Z0, …, Zn ) ∈ Pn | F0 ( Z0, …, Zn ) = …= Fmn( Z0, …, Zn ) = 0 }
Nonsingular connected projective manifold X V ( F0, …, Fm )
Homogeneous polynomial G1 ( Z0,… : Zn ), …, Gk ( Z0, … : Z n)
W = V ( F1, …, Fm, G1, …, Gk )
Algebraic cycle ∑iaiWi (Wi is irreducible submanifold. ai∈Z )
[References]
<Projective space>
Algebraic Linguistics / Linguistic Note / 7 Projective Space / Tokyo July 26, 2007
Tokyo November 19, 2007
Sekinan Research Field of Language
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