## 33.26 Types of varieties

Short section discussion some elementary global properties of varieties.

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.

Note that a projective variety is a proper variety, see Morphisms, Lemma 29.43.5. Also, an affine variety is quasi-projective as $\mathbf{A}^ n_ k$ is isomorphic to an open subscheme of $\mathbf{P}^ n_ k$, see Constructions, Lemma 27.13.3.

Lemma 33.26.2. Let $X$ be a proper variety over $k$. Then

1. $K = H^0(X, \mathcal{O}_ X)$ is a field which is a finite extension of the field $k$,

2. if $X$ is geometrically reduced, then $K/k$ is separable,

3. if $X$ is geometrically irreducible, then $K/k$ is purely inseparable,

4. if $X$ is geometrically integral, then $K = k$.

Proof. This is a special case of Lemma 33.9.3. $\square$

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