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!