Closed embeddings of manifold/varieties
In Italian, varietà means both manifold and variety. Here is one of the many theorems that exists for both, and which my coauthor and I have used recently in a work in progress. Theorem We have that f:X → Y is a closed embedding if and only if it satisfies all the following conditions: f is injective on points f is injective on tangent vectors f is closed (as a map of topological spaces) Algebraic version (AV): f is a morphism between schemes of finite type over an algebraically closed field k, we only consider k points. Differential version (DV): f is a smooth (holomorphic) map between differentiable (complex) manifolds (all manifolds are without boundary). We write k for R or C, respectively. A closed embedding means an isomorphism to a closed subscheme AV or submanifold DV of Y. Note that the conditions indicated are clearly necessary (only if). Proof AV Let Z be the schematic closure of the image. Since f is closed, f:X → Z is surjective; by 1. it is b...