Affine Algebraic Variety and Algebraic Variety

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Ref: Algebraic Geometry, Daniel Perrin.
We record definition of Affine Algebraic Variety and Algebraic Variety
  1. Harpreet Bedi
  2. Algebraic Geometry
  3. Affine Algebraic Variety
  4. Algebraic Variety
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But not all open sets of X are affine open sets. For example is X is k2. The point (0, 0) is a closed set in k2 defined by ideal (x, y) so k2 - (0, 0) is an open set of X. But this is not of the form D(f) for a single function f. It is not isomorphic to any affine algebraic set even in some higher dimension kn. However k2 - (0, 0) can be expressed as the union of two affine open sets of the form D(f), namely D(x) and D(y) which are themselves both isomorphic to some irreducible closed algebraic set in k3, namely V(J) where J= x + (tx-1) and J= y + (ty-1) respectively.

brucegreetham