# What is “embedded affine variety”

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:

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").

• As far as I understand, the term means that $X \in \mathbb{A}^n$ in a specific way for some $n$. For instance, $\mathbb{A}^1$ and $V(y-x^2) \subset \mathbb{A}^2$ are isomorhic, but the former is embedded in a line (and it is the line) and the latter is embedded in a plane. – Youngsu Jan 9 '18 at 12:49
"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).
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 $f(X)$.