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I tried to learn a bit about embeddings of algebraic varieties into other algebraic varieties. Depending on the textbook/online notes I consider, there are the notions of closed immersion and open immersion. Some of them say that the right notion of an "embedding" is a closed immersion but some say that one should rather take open immersion or even just immersion by which they mean a map that can be factored into an open immersion and a closed immersion.

1) What is the correct notion of "embedding" for a morphism $f:X\to Y$ between (smooth) projective varieties?
2) Can you explain the difference between open and closed immersions in an example?

Furthermore, whenever I read about explicit examples (e.g. projection from a point, Veronese embedding, Segre embedding,...), it is only shown that the map is injective and not a single word is said about what happens on tangent spaces. So the condition that actually turns an injective morphism into an embedding is never checked.

3) Why is that so? Under which conditions is it clear that we constructed an embedding?

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1) Correct, I don't know; the most general useful notion is probably the last one, where you can factor it into an open immersion followed by a closed one. This, for example, "immerses" the affine line $\mathbb A^1$ into the projective plane $\mathbb P^2$ via $x\mapsto (x:1)\in \mathbb P^1$ followed by $(x,y)\mapsto (x:y:0)\in \mathbb P^2$.

2) A closed immersion is like, for example, a line in the affine plane: It embeds the variety as a closed subvariety. The affine line into the projective line in 1) above is an example of an open immersion.

3) Injectivity is not sufficient to guarantee a closed immersion; see, for example, the map from the affine line to the cuspidal cubic $(t^2,t^3)$ in $\mathbb A^2$. The key thing is that pullback gives a surjective map of structure sheaves (coordinate rings in the affine case). It happens in the examples you give that this surjectivity holds.

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