What is the relation between an affine variety and an affine space?

Question

I was learning about algebraic varieties and the wikipedia page presents affine varieties as the "conceptually easiest" type. Having read about affine spaces and affine varieties, I am unsure of how the two concepts are connected.

As I understand it an $n$-dimensional affine space $\mathbb{A}^n$ over a field $K$ is the same set as the vector space $K^n$ only with some properties "ignored", i.e. there is no privileged origin point and elements may not be added unless the scalar coefficients sum to 0 or 1.

Now as far as I'm aware, an affine variety $V(S)$ for a set of polynomials $S\in K[x_1,\dots,x_n]$ is $$ V(S) = \{ (a_1,\dots,a_n),\; a_i\in K\;\; |\;\; f(a_1,\dots,a_n)=0\;\; \forall f \in S \}. $$ The (admittedly introductory) sources I've read describe the elements of this set as being points in an affine space but seem to effectively treat them as just tuples of $K$-values. Indeed you can associate them to points of an affine space, even points of a vector space if you set $\mathbf{0}=(0,\dots,0)$ but I'm curious as to the motivation for this.

Why is it important that the elements of $V(S)$ be a subset of a set with the structure of an affine space? Is anything lost by ignoring this structure? Conversely, is anything lost if we impose further structure and just say that they're part of a vector space?

Answer 1

As you've already noted, there is no distinguished origin in an affine space. This means, for example, that you don't have a notion of the sum of two points, or a scalar multiple of a point.

On the other hand, in algebraic geometry, you often want to put a topology on affine space. Here's a quote from An Invitation to Algebraic Geometry by Smith, Kahanpaa, Kekalainen, and Traves (who work over $\mathbb{C}^n$ mostly, rather than a general field):

To emphasize the difference between the vector space $\mathbb{C}^n$ and the set $\mathbb{C}^n$ considered as a topological space with its Zariski topology, we will denote the topological space by $\mathbb{A}^n$, and call it affine n-space. In particular, there is no distinguished "origin" in $\mathbb{A}^n$.

Why is it important to distinguish between the same set but with different structures? This is partly due to the influence of category theory, and also maybe model theory. Category theory was born partly from the realization that when studying structures, you should study the appropriate maps (morphisms) between them at the same time. For vector spaces, these are the linear maps; these preserve the origin and the vector space operations. For affine varieties, these are the polynomial maps. Very different morphisms for the two different categories.

In model theory, a structure is a domain (or universe) plus relations, constants, and functions on the domain. So here again we need to distinguish.