I have something that has been bothering me for a while: The concept of divisor. Let X be an affine variety, let's say smooth. Let $E_i $ be a prime divisor (i.e of codim 1). Let's $ D= \sum a_i E_i $.
My questions are:
- is $E_i $ a point?
- What's D? Is it a curve, a plane?
- The sum of these $E_i $, how it defined? Is it adding points in affine space?
- Can you please give me an explicit easy example so I can see what's going on?
Thank you.
I'm assuming you're working with Weil divisors (maybe they were just introduced to you as divisors). These are, by definition, finite $\mathbf{Z}$-linear combinations of closed subvarieties of codimension $1$. If $X$ has dimension $1$, then indeed such ``prime divisors" are points, but if $X$ has dimension larger than $1$, prime divisors will not be points. A point in a variety of dimension $n$ has codimension $n$, so prime divisors in a variety of dimension $n$ will be of dimension $n-1$.
As for your second question, $D$ is a formal object (an element in the free abelian group on the prime divisors of $X$), but if $D$ is effective in the sense that the non-zero coefficients $a_i$ are positive, then there is a geometric object associated to $D$, at least if you broaden your framework to that of schemes (I'm not really sure how this works from the "classical viewpoint"). For a general scheme, effective Cartier divisors are closed subschemes whose ideal sheaf is invertible, meaning, on some affine open covering, is principal, generated by a regular element (an element which isn't a zero divisor). When $X$ is sufficiently nice, there is a connection between invertible sheaves on $X$ and Weil divisors (depending on how you develop the theory, it goes through the intermediate notion of Cartier divisor, a concept which has ostensibly caused confusion for many great algebraic geometers over the years). You could talk about the schematic support of a Weil divisor as the scheme-theoretic union of the prime divisors occurring with non-zero coefficients. Anyway, again, the theory is probably most robust if you work with schemes instead of varieties in the sense of e.g. the first chapter of Hartshorne.
For your third question, the sum is formal. I guess geometrically you could imagine it as the union of the prime divisors occurring with non-zero coefficients and ``multiplicities" given by the coefficients $a_i$, but, in the classical framework, I don't know how valuable this is. If you work with effective Cartier divisors on schemes, you can indeed add them by taking the subscheme defined locally by the product of defining equations for the two divisors. This does give precise geometric meaning to the formal sum (again, in the case of non-negative coefficients).
For some examples, I'm not sure what you're looking for exactly. You can take any variety you want are start making integer linear combinations of prime divisors. For instance, you could take $X=\mathbf{A}_k^1$ over an algebraically closed field $k$. Then prime divisors are points, so a divisor is a formal integer linear combination $\sum_x a_x\cdot [x]$ with the sum is over the points $x$ of $X$ (closed points if you regard $X$ as a scheme) and the integers $a_x$ are almost all zero. Each (closed) point $x$ constitutes an effective Cartier divisor. Likewise for $X=\mathbf{P}_k^1$. For $X=\mathbf{A}_k^2$, given by codimension one subvarieties, which are all cut out by just one non-constant irreducible polynomial, i.e., by an equation $f(x,y)=0$ for some non-constant $f$. So a typical Weil divisor has the form $\sum_{i=1}^m a_m\cdot[V(f_i)]$, where the $f_i$ are irreducible polynomials in $k[x,y]$.
I'm not sure how useful this answer is, and there are certainly users on MSE with more knowledge who can furnish you with much more insight. Hopefully one of them will come along and provide a really helpful answer.
P.S. For what it's worth, I prefer to think of Weil divisors in the context in which they work best (integral Noetherian schemes which are regular in codimension $1$, or, classically, nonsingular varieties). They behave well on such schemes and strike me as inherently geometric. For general schemes, I like to think just of effective Cartier divisors (closed subschemes with invertible ideal sheaf). I've personally always found general (not necessarily effective) Cartier divisors on a scheme $X$ as defined in the standard expository textbooks (as global sections of a certain sheaf $\mathscr{K}_X^\times/\mathscr{O}_X^\times$ which is quite perilous to define for non-integral schemes) to be confusing. I would rather just think about invertible sheaves, which is where this last notion meets the "schematic" notion of an effective Cartier divisor. But this is merely my opinion, informed by my personal experience in learning this material and (especially) my intellectual limitations (which are considerable).