I've seen divisors on curves before, a few years ago in a course in algebraic geometry. Now I've come across them again, but they're somewhat more generalized. I was hoping someone could explain the similarities/differences between the two notions.
The first notion (from Algebraic Curves by Fulton) says for any irreducible projective curve $C$, with nonsingular model $X$, a divisor on $X$ is a formal sum $ D = \sum_{P\in X} n_P P$ where each $n_P\in \mathbb{Z}$ and all but finitely many are 0.
The second notion (from Algebraic Geometry by Milne) says for any normal, irreducible variety $V$, a divisor on $V$ can be written uniquely as a finite (formal) sum $ D = \sum n_i Z_i,$ where the $Z_i$ are the irreducible subvarieties of $V$ of codimension 1 (prime divisors).
In the generalized case, when $V$ is a curve (i.e., dimension 1) then the $Z_i$ are 0 dimensional, so just points on the curve, and everything seems OK. But then we move on to the degree of a divisor. In Fulton, the degree of a divisor $\sum_{P\in X} n_P P$ is simply the sum of the coefficients, $\sum_{P\in X} n_P$. However, via Example 12.4 in Milne, the definition of a divisor is as follows:
Let $C$ be a curve. If $D = \sum n_i P_i$, then the intersection number $ (D) = \sum n_i[k(P_i):k].$ By definition, this is the degree of $D$.
Now, we don't have any references for my class (as it's all over the place and would require about 10 minimum..), but this is how we defined the degree of a divisor for a curve in class (using the extension of $k(P_i)$ over $k$) and I'm confused. Is this suggesting that, by Fulton's definition, we have $[k(P_i):k] = 1$ for every point? Because if so, it seems silly to even bother putting that in. Now given Milne is actually writing the intersection number, I felt he could just leaving implicit that the $[k(P_i):k]$ are 1. However in class, we simply said a divisor is blah and its degree is blahh using the degree of the extensions, so there must be something non-trivial going on here!
Now the only thing I haven't entirely accounted for above, is the fact that in Fulton, a curve over a field $k$ is a set of points in $k^n$, whereas Milne is using algebraic varieties, which I'm not terribly well versed with. However, Milne makes note of the fact that there is a one-to-one correspondence between maximal ideals of $k[V]$ and one point sets of $V$, so I would expect, for each $P_i$ in Fulton's definition, there is a corresponding maximal ideal, so that a divisor should be identical, except replacing each $P_i$ with the corresponding maximal ideal.
Somewhat more troubling, I thought it was obvious at first, and didn't give it a second thought, but in retrospect, what is $[k(P_i):k]$? If $P_i$ is a subvariety of codimension 1, for a curve defined over $k$, then how can this not have to be 1? If nothing else, I'd also appreciate any additional references for these generalized divisors on curves (though I do like Milne's notes!). Thanks!