Embedding an affine variety in affine space So in Hartshorne's Algebraic Geometry, chapter 1 sections 4 and 5 he mentions how 2 definitions (the blowing-up of a variety at a point, and a point being non-singular of affine varieties) "apparently depend upon the embedding of the $Y$ in $\mathbb A^n$".
What does this actually mean?
In the 'non-singular point' definition (Let $f_1,...f_n$ be a set of generators of $I(Y)$, then $Y$ is non-singular at $p$ if the rank of the matrix $(\partial f_i / \partial x_j)(p)$ is $n- dim(Y)$)  he mentions that it is easily checked that $Y$ being non-singular at $p$ is independent of the generators of $I(Y)$ chosen, so I am unsure what else he could mean by being 'dependent upon the embedding of $Y$ in affine space'.
 A: What he's trying to say is this: it's not immediately clear that these properties are conserved under isomorphisms.
To be more explicit, let $\phi:Y\to X$ be an isomorphism of varieties, where $X\subseteq \Bbb A^m$. It's it the case that $p\in Y$ is singular iff $\phi(p)\in X$ is singular? If we blow up $Y$ at $p$, it the result isomorphic to what we get if we blow up $X$ at $\phi(p)$?
A: Apparently is the crucial word. The definitions seem like they might depend on the embedding, but they do not. (Though you have to prove it.) 
Let's think of this algebraically. Imagine you define some properties of a k algebra in terms of a presentation (generators and ideal of relations - which is the same thing as an embedding into affine space), perhaps in terms of explicit computation with the equations as is done for non singularity. Then you need to make sure it is independent of the presentation you chose in order for it to be a property of the ring. Otherwise , it will be a property of the presentation. 
You've seen this with groups, probably. The number of generators is not a property of the group, but the minimal number of generators is. Similarly, the number of generators is not a property of the ring, but the minimal number of generators is. (The smallest affine space it can be embedded in.)
If you look at the definition of non singularity, since you are making computations with the presentation of the ring, it may seem that it depends on the presentation. Later, when you learn the definition of non singularity in terms of the local ring, you will see that this definition with derivatives is better thought of as a technique for computing a certain invariant.
