What is “embedded affine variety”

Question

I am a junior learner of algebraic geometry. And I just roughly go through the following lecture:

https://math.berkeley.edu/~brandtm/talks/sonberkeley.pdf

In the first page, Definition 2. says:

If $X$ is an embedded affine variety, then its projective closure $\bar{X}$ is the smallest projective variety containing $X$.

So my first question is what is "embedded affine variety". So far I cannot find its definition in my algebraic geometry textbook and the internet.

Moreover, I also read the following discussion:

algebraic/geometric interpretation of the projective closure of an affine variety

It seems that in the above definition, the "embedded" term is not necessary. (Please see the answer of this discussion, the article did not mention "embedded").

I am confused about these issues. Please advise, thanks!

Answer 1

"Embedded affine variety" means an affine variety with a chosen embedding into some affine space $\Bbb A^n$. Since the embedding $\Bbb A^n\hookrightarrow\Bbb P^n=\operatorname{Proj} R[x_0,\cdots,x_n]$ given by identifying $\Bbb A^n$ and $D(x_0)$ gives you a map from any variety $X\subset \Bbb A^n$ into $\Bbb P^n$, you can then talk about the closure of it's image, which is what is meant by "the smallest variety containing $X$".

It should be noted that in the discussion you link, the variety comes equipped with a chosen embedding - the data of $I\subset k[x_1,\cdots,x_n]$ determines how the variety is embedded. An embedding of a variety $X=\operatorname{Spec} k[X]$ is equivalent to a choice of generators for $k[X]$ as a $k$-algebra, and it is not immediately obvious that different choices should produce the same projective closures (for example, I'm not even sure that this is true - it could be an interesting question to think about for a little bit).

Answer 2

The term "embedding" is somewhat ambiguous in algebraic geometry. People typically use the word to mean that a map $f: X \to Y$ of schemes or varieties is a closed immersion. In precise terms, a closed immersion is a map $f$ on the underlying topological spaces that identifies $X$ with $f(X)$, a closed subset of $Y$, together with a surjection of sheaves $\mathcal{O}_Y \to f_*\mathcal{O}_X$. Alternately, one might use the word to mean open immersion, or a map $f$ that's a homeomorphism onto its image and that $\mathcal{O}_Y \to f_*\mathcal{O}_X$ restricts to an isomorphism on $f(X)$.