The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Definition 32.26.1. 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 $X$ it 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 28.41.4 this is true if and only if $X$ is isomorphic to a closed subscheme of $\mathbf{P}^ n_ k$ for some $n$.

  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 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 $n$.

  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.


Comments (1)

Comment #3778 by Tim Holzschuh on

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


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