The Stacks project

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

1. 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 $n$.

2. 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 $n$.

3. 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 $n$.

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

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