1 n-dimensional complex projective space Pn is defined by the following.
Sch is category of scheme. Set is category of set. X is scheme.
Contravariant functor hx : Sch → Set
hx (S) = Homsch (S, X)
Pn = hx (Spec(C))
Pn becomes scheme for expressing hx.
2 P2 that is defined by 2-dimensional homogeneous polynomial is presented.
Closed scheme of P2, X = Proj C |x, y, z| becomes <line>, <circle> or <surface> by affine transformation.
<Surface> is projective manifold.
3 Weighted projective space that is projective algebraic manifold is defined by the following.
P (a0, …, an) = Proj C |x0, …, xn|
4 Weight is identificated to word order of language.
5 Language is expressed by algebraic manifold, especially by projective scheme.
[Reference]
Tokyo June 11, 2007
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