We say $X$ is an affine variety if $X$ is an affine scheme. This is equivalent to requiring $X$ it be isomorphic to a closed subscheme of $\mathbf{A}^ n_ k$ for some $n$.

We say $X$ is a projective variety if the structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is projective. By Morphisms, Lemma 28.41.4 this is true if and only if $X$ is isomorphic to a closed subscheme of $\mathbf{P}^ n_ k$ for some $n$.

We say $X$ is a quasi-projective variety if the structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is quasi-projective. By Morphisms, Lemma 28.38.6 this is true if and only if $X$ is isomorphic to a locally closed subscheme of $\mathbf{P}^ n_ k$ for some $n$.

A proper variety is a variety such that the morphism $X \to \mathop{\mathrm{Spec}}(k)$ is proper.

A smooth variety is a variety such that the morphism $X \to \mathop{\mathrm{Spec}}(k)$ is smooth.

In there is a little typo: "This is equivalent to requiring X be isomorphic [...]"

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## Comments (1)

Comment #3778 by Tim Holzschuh on