I am reading course notes on algebraic geometry, where a morphism of varieties is defined as follows ($k$ is an algebraically closed field):
Let $X$ be a quasi-affine or quasi-projective $k$-variety, and let $Y$ be a quasi-affine or quasi-projective $k$-variety. A map $f:X\to Y$ is called a morphism of $k$-varieties if $f$ is continuous, and if for every open subvariety $U$ of $Y$ and every regular function $f:U\to k$ the composition $h\circ f$ is a regular function on $f^{-1}(U)$.
I have trouble seeing the motivation for this definition.
The above notion of morphism seems to imply that the 'structure' of a variety (what distinguishes it from a mere set) is the following:
- A topology, the Zariski topology, which is extremely coarse (weak) compared to the Euclidean topology in the case $k=\mathbb{C}$.
- For every Zariski-open $U\subseteq X$ a specification of which functions $U\to k$ are considered 'nice', i.e. regular.
Given that this is indeed the structure we want to assign to a variety, I agree with the above notion of morphism. In particular, I can see that two varieties are isomorphic exactly when their structure is 'the same' (so that the difference between them is just that their points have different names).
However, I fail to see why the above two bullets accurately capture what we want them to. In fact, I do not know when we want to consider two varieties as isomorphic, and why. I do not know why we want the curves defined by $x^2+y^2=1$ and $x^2+y^2=2$ to be deemed isomorphic, but not the affine plane and the punctured affine plane (except for just pointing back to the definition I am trying to motivate, and showing that no isomorphism exists algebraically, but that's not enlightening). I do know this in the category of smooth manifolds: I expect two spheres to be diffeomorphic because I can stretch one smoothly to exactly match the other. I expect a sphere and a torus not to be diffeomorphic because no matter how hard I try, I cannot strech the sphere and make it coincide with a torus.
Another example: the affine line and the cusp are not isomorphic, and the difference lies exactly in the singularity of the cusp (its… well, cusp). Is this what we want to encode, the behaviour of varieties near singularities (I suspect this is only part of what we want to encode)? Do we want two varieties to be isomorphic if there is a bicontinuous bijection between them that maps singularities to singularities of the same kind? (Here, I do not know what I mean by the 'kind' of a singularity, and in fact I don't even know what exactly I mean by a singularity.) I expect that isomorphic varieties will have 'analogous' singularities at corresponding points, but I suppose there is more to the structure of a variety than this (indeed, not all smooth varieties are isomorphic).
What do we want the structure of a variety to entail intuitively? What intuitive/geometric information is encoded in the 'structure' as outlined in the bullets above?
Edit. I want to know what is encoded in the structure of a variety on a vague and intuitive level. I do not require mathematical justification for the answers at all (no need to prove that this is what we encode).
Best Answer
Mariano Suarez-Alvarez's point about understanding the intuition as you learn the theory more is correct, but I'd like to give a partial answer to help guide your intuition. After all, it is possible to spend months or years learning algebraic geometry and come away with little intuition of what the whole subject is about.
First, algebraic varieties are geometric spaces which look locally like affine varieties. In this sense, the theory is developed similar to, say, the theory of manifolds where a manifold is defined to be a space that is locally Euclidean. Of course, that limits the local study of manifolds - any two manifolds are locally isomorphic. Not so for algebraic varieties, as there is a wide variety of affine varieties.
So I think you should begun by restricting your question to affine varieties. And the key is that affine varieties are completely determined by their ring of globally regular functions. In other words, two (irreducible) closed subsets of affine space are isomorphic iff we can find a global 'change of variables' that identifies the global regular functions on the two spaces. Rescaling $(x,y) \mapsto (\sqrt{2}x,\sqrt{2}y)$ yields the isomorphism between $x^2+y^2=1$ and $x^2+y^2=2$.
I'll modify your non-example (because $\mathbb{A}^2 \setminus \{0\}$ is not affine) and explain why $\mathbb{A}^1$ and $\mathbb{A}^1 \setminus \{0\}$ are not isomorphic. Their rings of regular functions are $k[T]$ and $k[T,T^{-1}]$ respectively, which are not isomorphic. So there can be no 'changes of variables' that identifies the two spaces.
One important caveat: when I say there is global 'change of variables' from $X \subset \mathbb{A}^n$ and $X' \subset \mathbb{A}^{n'}$, I am talking about using polynomial maps that are restricted from the respective affine spaces, but they only needed to be defined on the spaces $X$ and $X'$. For example $\mathbb{A}^1 \setminus \{0\}$ (viewed as $t \neq 0$) and $xy=1$ are isomorphic via $t \mapsto (t, 1/t)$ and $(x,y) \mapsto x$. Of course, $1/t$ is only a valid change of variables when $t \neq 0$, but fortunately we are only looking at points where $t \neq 0$.
The global story is similar, except that we cannot just compare globally regular functions. (For example, the only globally regular functions on any projective variety are the constant functions, yet intuitvely there ought to be many different projective varieties up to isomorphism.) So now we require a global 'change of variables' so that regular functions on local pieces match up with the regular functions on the corresponding local pieces.
I am not sure if this explanation is what you are looking for. Algebraic geometry is very much a function oriented theory. We compare spaces by looking at the functions on them. One can take such an approach to manifolds as well. But for manifolds we also have an intuition for what the possible change of variables are ('stretching' and 'twisting' and the like). It's much harder to tell such a story in algebraic geometry because algebraic varieties are so much more diverse. There are still some basic intutions such as you can't have an isomorphism between a smooth variety and a singular variety because isomorphisms give rise to (vector space) isomorphisms of tangent spaces. But there are lots of possible singularities, and getting a hold on them is a major on-going project in the field. For example, you could study plane curves in depth and learn to tell apart singularities in this case (using blowups). But then you'll quickly discover the singularities on surfaces are more complicated and those on higher dimensional varieties still more complicated and hard to get a handle on.