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complete
A complete variety (integral separated scheme of finite fields) is a proper variety viewed as a \(k{\hbox{-}}\)scheme, so separated, finite type, and universally closed.
Idea: analog of compactness, e.g. \({\mathbf{A}}^n\) is not complete but \({\mathbf{P}}^n\) is. Projective implies complete for varieties over a field, and it’s hard to produce complete varieties which are not projective. A finite-type scheme \(X\) is complete (i.e. proper) over \({\mathbf{C}}\) iff \(X({\mathbf{C}})\) is compact Hausdorff.
Idea for universally closed: a variety is complete when every projection with \(X\) as a fiber is a closed map. Equivalently, every map \(X\to {\operatorname{pt}}\) is universally closed, i.e. the structure morphism is proper.
Idea: completeness wants to be an algebraic version of compactness, and properness wants to be a relative version of completeness.
Idea: you can’t miss limit points in \(X\) – but \(X\) itself couldn’t intrinsically know that. So “embed” \(X\) in a bigger space. How do you know if it’s big enough? Allow an arbitrary \(Z\) to tell \(X\) that it’s missing limit points by having something with a limit in \(Z\) come from something in \(X\times Z\) without a limit, i.e. the projection \(\pi:X\times Z\to Z\) is not closed.
How to check: the valuative criterion of properness.
Completeness for a toric variety: \(\mathop{\mathrm{supp}}\Delta = {\mathbb{E}}^n \coloneqq N_{\mathbf{R}}\), so the fan covers the ambient Euclidean space.
An affine complete variety over \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) is necessarily finite.
Example: \({\mathbf{A}}^1\) is not complete, since \(V(xy-1) \to {\mathbf{A}}^1\) projecting onto the \(x{\hbox{-}}\)axis is not closed, since its image is \({\mathbf{G}}_m = {\mathbf{A}}^1\setminus\left\{{ {\operatorname{pt}} }\right\}\).