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remark. affine v.s. projective [michalek2021invitation, ch. 2] [ag-000E]

The prefix "affine" of affine variety is used to distinguish it from a projective variety. Affine varieties arise from arbitrary polynomials, while projective varieties arise from systems of homogeneous polynomials, i.e. linear combinations of monomials of fixed degree.

Since affine varieties are the general case, they are sometimes simply called varieties.

Figure out its relation to affine space in Geometry, which preserves parallelism and ratio of lengths for parallel line segments, but not distances and measures of angles.