Divisors of a function

Question

What is a divisor of a function?

Let $X$ be an algebraic variety over $\mathbb{k}$, $f\in \mathbb{k}(X)$. There exists a way to build an integer $\nu_C(f)$ for each irreducible closed subset $C$ of codimension $1$. If $X$ is a curve then $C$ is just a point of $X$ and $\nu_C(f)$ is an order of a zero or a pole. I cannot understand the general case. Could you you give me a reference to this construction or even explain if it is not complicated in reality?

Answer 1

Suppose that X is sufficiently nice (integral, separated, normal for example). Then the divisor is $\displaystyle \sum_Y v_Y(f) Y$, where $Y$ runs over the irreduciblr closed codimension $1$ subvarieties of $X$, and $v_Y(f)$ is the valuation of $f$ along $Y$. This is defined by noting that since $Y$ is codim 1, if $\eta$ is its generic point, then $\mathcal{O}_{X,\eta}$ is a DVR. So there is an associated valuation $v_Y$ on $k(X)$. Then, $v_Y(f)$ is just the valuation of $f$.

Intuitively, $v_Y(f)$ is the order of a zero or pole 'along' $Y$. For example, in $ \mathbb{A}^2$, the y-axis is a codim 1 subset, and the function $\frac{1}{x^2}$ has a pole of order 2 along it.

Answer 2

$\newcommand{\oO}{\mathcal{O}}$

Firstly, note that what you're talking about can only be done for varieties which are regular in codimension 1, which is to say that the local ring of the generic point of every codimension 1 subvariety is a discrete valuation ring (DVR). In the case of curves, what this means is that you can only talk about functions which have neither zeros or poles on singular points of the curve, or if you want to take the divisor of any function, you want to only consider nonsingular curves.

In the case of curves, the number $\nu_C(f)$ is basically found as follows: Find some open affine neighborhood $U$ of the point $C$ with ring of regular functions $\oO_X(U)$. The function $f$ can then be thought of as an element of the field of fractions of $\oO_X(U) = k(X)$. Since $X$ is a curve, the ring $\oO_X(U)$ will have Krull dimension 1, and the point $C$ corresponds to a maximal ideal $\mathfrak{p}$ of this ring of height 1. The local ring at $C$ is then just the localization $\oO_X(U)_\mathfrak{p}$. Assuming $X$ is regular in codimension 1, this means that the local ring $\oO_X(U)_\mathfrak{p}$ is a discrete valuation ring, which has associated to it a discrete valuation, namely your $\nu_C$. Basically, this is just the number of times a uniformizer (ie, a generator for the principal ideal $\mathfrak{p}\oO_X(U)_\mathfrak{p}$) divides $f$.

For any function $f$, the number $\nu_C(f)$ should be thought of as the order of zero (or pole) of $f$ at $C$. This is similar to the notion of zeroes and poles you study in complex analysis. However, most 1st year complex analysis generally only consider Riemann surfaces, ie the geometry of 1-dimensional complex varieties. In higher dimensions, you can no longer talk about the zero of a function at a (closed, ie, non-generic) point. This is because when you look at the local ring of a closed point, the local ring has dimension > 1, and it's impossible for its maximal ideal to be generated by a single element, so you can't define the order of $f$ at that point as a single number anymore. Instead, you have to talk about orders of zeros and poles along codimension 1 closed subvarieties.

The process described for curves generalizes easily to higher dimensional varieties which are also regular in codimension 1. In other words, for any codimension 1 subvariety $C$, the local ring of its generic point will be a DVR with fraction field $k(X)$, and so you can talk about zeros and poles of functions at $C$ in exactly the same way. If you don't know what generic points are, you can rephrase the process this way. Let $C$ be your codim 1 subvariety, and let $U$ be any affine open set of $X$ intersecting $C$ nontrivially. Then the ring of regular functions $\oO_X(U)$ on $U$ is an integral domain, and the codimension 1 subvariety $C$ cuts out a codimension 1 subvariety of $U$, corresponding to a height 1 prime $\mathfrak{p}$ of $\oO_X(U)$. The local ring at the generic point of $C$ is then just $\oO_X(U)_\mathfrak{p}$.

The divisor of a function $f$ in the function field is then just the formal sum $$\sum_{C\subset X\text{ codim 1}}\nu_C(f)\cdot C$$