Definition 33.26.1. Let $k$ be a field. Let $X$ be a variety over $k$.

We say $X$ is an

*affine variety*if $X$ is an affine scheme. This is equivalent to requiring $X$ to 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 29.43.4 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 29.40.6 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.

## Comments (2)

Comment #3778 by Tim Holzschuh on

Comment #3908 by Johan on