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I am following Fulton's book on Algebraic curves.It starts with the notion of affine spaces denoted by $\mathbb A^n(k)$ where $k$ is a field.The author defines the affine space $\mathbb A^n(k)=k^n$ the $n$-fold cartesian product of the field $k$ and there is no additional structure on it.So,my question is about the notation.Why cannot we simply denote the affine space by $k^n$.Why we denote it with a more complicated notation $\mathbb A^n(k)$?Algebraic curves is almost new topic to me and I have no idea why this is defined by such a notation.Can someone clarify?

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If you write $k^n$ it will most likely be interpreted as the $n$-dimensional vector space over $k$. The $n$-dimensional affine space over $k$ is a different object, it’s not a vector space, and although both objects can be identified as sets they belong to different categories and have different morphisms (affine space is commonly thought as a scheme). So the notation $\mathbb{A^n}(k)$ is preferred because it is less ambiguos, and it is consistent with the notation $\mathbb{P}^n(k)$ for projective space.

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Yes, $\mathbb A^n(k)$ is just $k^n$ but Fulton's motivation for the notation is probably to distinguish the set of $n-$tuples of members of $k$ from the projective space $\mathbb P^n(k)$, which is $k^{n+1}\(0, ... ,0)$ modulo the equivalence relation of scalar multipication by a non-zero member of $k$.

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