linguistic:paraphrasing, linguistic:translation, linguistic:voice_change, structural:expansion
Non-examples

Consider the affine line A 1 {\displaystyle \mathbb {A} ^{1}} over C {\displaystyle \mathbb {C} }. The complement of the circle { z ∈ C with | z | 2 = 1 } {\displaystyle \{z\in \mathbb {C} {\text{ with }}|z|^{2}=1\}} in A 1 = C {\displaystyle \mathbb {A} ^{1}=\mathbb {C} } is not an algebraic variety (nor even an algebraic set). Note that | z | 2 − 1 {\displaystyle |z|^{2}-1} is not a polynomial in z {\displaystyle z} (although it is a polynomial in the real cooridnates x, y {\displaystyle x,y}). On the other hand, the complement of the origin in A 1 = C {\displaystyle \mathbb {A} ^{1}=\mathbb {C} } is an algebraic (affine) variety, since the origin is the zero-locus of z {\displaystyle z}. This may be explained as follows: the affine line has dimension one and so any subvariety of it other than itself must have strictly less dimension; namely, zero.

For similar reasons, a unitary group (over the complex numbers) is not an algebraic variety, while the special linear group SL n (C) {\displaystyle \operatorname {SL} _{n}(\mathbb {C})} is a closed subvariety of GL n (C) {\displaystyle \operatorname {GL} _{n}(\mathbb {C})}, the zero-locus of det − 1 {\displaystyle \det -1}. (Over a different base field, a unitary group can however be given a structure of a variety.)