Some people might say that the natural place for this topic is before talk of differential forms and of the canonical bundle, but I disagree. Well, really it’s fine either way, but this is my blog, so I’m going to do it my way. Today we’re going to talk about divisors and their relation to line bundles.

First off, we need to define the local ring of a subvariety, because otherwise we’re restricted to looking at actually nonsingular varieties, when we can in fact get away with a little bit less for the nice version of divisors. If $Y\subset X$ is a subvariety, we define $\mathscr{O}_{Y,X}$ to be equivalence classes of rational functions on $X$ where $f$ is defined somewhere on $Y$. Note that the dimension, as a ring, of $\mathscr{O}_{Y,X}$ will just be the codimension of $Y$ in $X$. (For our purposes, $\mathrm{codim}(Y;X)=\dim X-\dim Y$)

So we say that a variety is regular in codimension one if the local rings for every codimension one subvariety are regular. That is, for each of these rings, the maximal ideal (of functions vanishing on $Y$) satisfies $\dim \mathfrak{m}/\mathfrak{m}^2=1$. Roughly what this means is that our variety isn’t singular along codimension one subvarieties, but it may be for higher codimension (that is, lower dimension). So a curve has to be nonsingular, but a surface can be singular at a point, and a threefold can even be singular along a curve.

So now, we define a prime divisor to be a subvariety of codimension one (thus the need for regularity). In general, a Weil Divisor is a formal integer linear sum of prime divisors. So we act like we can add and subtract subvarieties. We can write any divisor as $D=\sum n_i Y_i$ where $Y_i$ are prime divisors and $n_i$ integers. We call a divisor effective if the $n_i$ are all non-negative.

So now take an arbitrary rational function $f$ on $X$ which isn’t identically zero. We want to define the order of vanishing of $f$ along the prime divisor $Y$. We do this by looking at the image of $f$ in $\mathscr{O}_{Y,X}$. It is a theorem that we can write $f$ as $ut^n$ for the generator of the maximal ideal $t$. We define $v_Y(f)=n$, in this case. It happens that there are only finitely many subvarieties such that $v_Y(f)$ is nonzero. This will always be a positive number, and we call it the order of the zero along $Y$. Now, for those subvarieties on which $f$ can’t be defined, we look at $1/f$, which can be extended to a function that is zero along them, and define $v_Y(f)$ to be $-v_Y(1/f)$ along them, and call it the order of the pole along $Y$.

We call any divisor of the form $(f)=\sum v_Y(f)Y$ a principal divisor. We note that $(f/g)=(f)-(g)$, and so taking a function to its divisor gives a homomorphism $K^*\to \mathrm{Div} X$, the group of Weil divisors on $X$.

So now a couple of really important definitions: we say that two divisors are linearly equivalent if $D-D'$ is principal, and we define $\mathrm{Cl}(X)=\mathrm{Div}(X)/\mathrm{Princ}(X)$.

Now, Weil divisors are actually not terribly good, but that’s how we’ll often be speaking of divisors, when we use them (mostly like points on a curve of curves in a surface or the like). What we REALLY want are called Cartier Divisors and for these we fundamentally need sheaf theory.

Denote by $\mathscr{K}$ the constant sheaf taking values the field of rational functions on $X$, and then $\mathscr{K}^*$ is the group of invertible elements. Now $\mathscr{O}^*$ is the sheaf of invertible regular functions. We look at the sheaf $\mathscr{K}^*/\mathscr{O}^*$, and note that there is a sheafification that needs to be performed to get here. We define a Cartier divisor to just be a global section of this sheaf. We call a Cartier divisor principal if it is in the image of the natural map $\Gamma(X,\mathscr{K}^*)\to\Gamma(X,\mathscr{K}^*/\mathscr{O}^*)$.

Now, a Cartier divisor can be written as an open cover of $X$ and on each element $U_i$ of the cover, a nonzero element of the function field $f_i$ such that on $U_{ij}$ we have $f_i/f_j$ a regular function. Though this is a multiplicative group, we will speak as though it is additive, because we really want to pretend that these are Weil divisors. So we define a pair of divisors to be linearly equivalent if their difference (ratio) is prinicpal.

Though it isn’t in general the case, for nonsingular varieties, the notions of Weil and Cartier Divisors agree. More generally, we’ll always use Cartier divisors.

So now, let $D$ be a (Cartier) divisor. We can use this to define a Line Bundle! We do this by taking the sub-$\mathscr{O}_X$-module of $\mathscr{K}$ generated by $f_i^{-1}$ on $U_i$. Because $f_i/f_j$ is invertible on the overlap, the sheaves we get will coincide, and so we get a sheaf on the whole space. We call this line bundle $\mathscr{L}(D)$.

The nicest fact, is that this gives a homomorphism from Cartier divisors modulo linear equivalence to the Picard group! So linearly equivalent divisors define the same line bundle, and the group operations are compatible. Even better, the homomorphism is injective. It isn’t always surjective, however, over the more general spaces often considered (like schemes), but is is true for varieties by our definition of them.

Just a little bit more, and then we’re done for the day. We call a Cartier divisor effective if it can be represented by regular functions on the open sets. Then it defines a subvariety, which then has ideal sheaf $\mathscr{I}_Y$. We want to be able to say that $\mathscr{I}_Y$ is isomorphic to $\mathscr{L}(-D)$, but there is one problem. Divisors can have multiplicity along subvarieties. So if the regular functions vanish only to order one, then we’re fine. Otherwise, we’ll need a more general notion of a subvariety, which I would want to call a nonreduced variety, but we’re not going to get into this (at least not yet, they’re easiest to describe as schemes, but I’m trying to avoid that word).

And as we finish, one last definition. A divisor corresponding to the canonical bundle will be called a canonical divisor, and all canonical divisors are linearly equivalent. We will most definitely use this in the future.