Philosophy behind effective Cartier Divisors

Question

I learned that effective Cartier Divisors provide a nice tool for anaysing cohomology groups:

If we have a curve $C$ and an invertible sheaf $\mathcal{L} \in Pic(C)$, we can use a Cartier Divisor $D$ with corresponding global section $s \in \Gamma(C, \mathcal{L})$ to get an short exact sequence $$ 0 \to \mathcal{O}_L \xrightarrow{\text{s}} \mathcal{L} \to \mathcal{O}_D \to 0 $$

which induces long exact sequence

$$ 0 \to H^0(\mathcal{O}_L) \xrightarrow{\text{s}} H^0(\mathcal{L}) \to H^0(\mathcal{O}_D) \to H^1(\mathcal{O}_L) \to ...$$

If there are given some extra informations about cohomology groups of $\mathcal{L}$, we can get new informations about cohomology groups of tzhe struructure sheaf $\mathcal{O}_L$.

But I suppose that there is a much more deeper philosophy about application / understanding of effective cartier divisors. What is it's geometrical interpretation / motivation? Can anybody explain the intuition behind this?

Answer 1

The origin

Clearly [divisor] comes from the term [divisibility].

For example we say 3 is a divisor of 6 when $3|6$. In $\mathbb{N}$, we have the beautiful result the prime factorization theorem:

$\forall n\in \mathbb{N},$ there exists a unique prime factorization $n=\prod_i p_i^{n_i}$.

i.e. $n$ can be uniquely determined by the formal sum $\sum_i n_i \cdot [p_i]$.

This property is also called unique factorization. The rings with this property are call unique factorization domains. It's a nice property but it's a bit strong, i.e. this property doesn't cover enough rings that we care about.

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Generalization in algebra

We discover another property that is slightly weaker than unique factorization so it is still useful and it covers more rings.

This is called: integral domains with a divisor theory, see this link: Basically if $A$ is an integral domain, a divisor theory of $A$ is a map of abelian semi-groups with units $A\backslash \{0\}\to D_0$ where $D_0$ has unique prime factorization, s.t. this map satisfies three more properties about divisibilities.

The good news is that every integral domain has at most one divisor theory. Note that $D_0$ must be a freely generated over $\mathbb{N}$. And the map naturally extends to the fraction field $K$ of $A$: $K^*\to D$ where $D$ is the free abelian group associated to $D_0$.

Now we can start to put some names:

1. Element of $D$ can be called divisor of $A$ or $K$. And $D$ can be also written as $\mathrm{\mathop{Div}}(A)$
2. Element of $D_0$ can be called effective divisor or non-negative divisor.
3. Generators of $D$ (or $D_0$) can be called prime divisors

We clearly care about the kernel, image and cokernel of the map $K^* \to D$

1. the kernel is simply the units of $A$, $A^*$
2. the element of image is called principal divisor, sometimes written as $\mathrm{\mathop{div}}(f)$. The image can be denoted as $\mathrm{\mathop{PDiv}}(A)$
3. the cokernel is then called the divisor class group of $A$ (or $K$), written as $C(A)$ or $\mathrm{\mathop{Cl}}(A)$.

For example, all Dedekind rings (usually appears in algebraic number theory) has a divisor theory.

So what's the point of a divisor theory?

Answer: We have the following exact sequence: $$0\to A^* \to K^* \to \mathrm{\mathop{Div}}(A) \to \mathrm{\mathop{Cl}}(A)\to 0$$ Clearly the divisor group $D$ has an extremely simple group structure. We want to use all the information above to obtain information about $A$ or $K$.

Generalization in geometry

When the idea generalize to a space $X$. We expect that:

1. the space $X$ has a well-defined function field $K(X)$.
2. Divisors should be freely generated by prime divisors so the structure of $\mathrm{\mathop{Div}}(X)$ is simple.
3. There exists a well defined map of groups $K(X)^* \to \mathrm{\mathop{Div}}(X)$ so we can relate $K(X)$ to $\mathrm{\mathop{Div}}(X)$, i.e. given a non-zero function on $X$, we can produce a divisor.
4. We want the above map to be as close to bijection as possible. So that we can deduce information about $K(X)$ from $\mathrm{\mathop{Div}}(X)$, then gain information about $X$ from $K(X)$.

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Next let's consider $A=\mathbb{C}[x]$ and $K=\mathbb{C}(x)$. Its prime divisor can be viewed as points of $\mathbb{C}$. So its divisors can be viewed as integral formal finite linear combination of points of $\mathbb{C}$.

And we found that principal divisor can be viewed as zeros/poles of rational functions in $\mathbb{C}(x)$.

Note that we are using divisor theory to define divisor.

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There are a few different ways to generalize here, because we can view $\mathbb{C}(x)$ as the function field of different type of spaces:

1. $\mathbb{C}(x)$ is the function field of $\mathrm{\mathop{Spec}}\ \mathbb{C}[x]$ as a scheme or algebraic variety.
2. $\mathbb{C}(x)$ is the function field of $\mathrm{\mathop{Proj}}\ \mathbb{C}[x,y]$ as a scheme or projective algebraic variety.
3. $\mathbb{C}(x)$ is the function field of $\mathbb{CP}^1$/Riemann sphere as a complex manifold.
4. $\mathbb{C}(x)$ is the function field of $\mathbb{CP}^1$/Riemann sphere as a complex analytic variety.

Usually we pick the third choice.

Q: Why not 1 or 2? A: When considering functions relating to complex analysis, we expect to consider holomorphic/meromorphic/analytic functions instead of polynomial/algebraic functions.

Q: Why not 4? A: the concept of complex manifold is more accessible than the concept of complex analytic variety. Though in higher dimension we should use complex analytic variety.

Q: In 3 why do we use $\mathbb{CP}^1$ instead of $\mathbb{C}$? A: By definition of divisor, we are not allowed to have inifinite sum, hence we can not allow a meromorphic function to have infinite number of zeros/poles. Over $\mathbb{C}$, a simple holomorphic function $\sin z$ has infinite zeros. So we can choose

1. Expand the group of divisor to contain infinite sum;

But it will make the group structure of $\mathrm{\mathop{Div}}(X)$ more complicated.

2. Only consider the functions that have finite number of zeros/poles.

Recall that we study the theorem of divisors to obtain information about $K(X)$ then deduce information about $X$. So if not all non-zero functions in $K(X)$ can properly generate a divisor, then there is not much point here.

3. Give up $\mathbb{C}$ and choose $\mathbb{PC}^1$ since non-zero meromorphic function on $\mathbb{PC}^1$ has finite number of zeros/poles.

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In short, now $X=\mathbb{CP}^1$, $K(X)=\mathbb{C}(x)$, prime divisors are points of $X$, divisors are freely generated by the prime divisors. Principal divisors are still the zeros/poles of a non-zero function in $K(X)$.

Next, let $X$ be a Riemann surface, i.e. a compact irreducible 1-dimensional complex manifold.

Irreducibility is to make sure the existence of function field $K(X)$.

Compactness is to make sure every non-zero function in $K(X)$ can generate a well defined divisor.

Being 1-dimensional complex manifold is just copying from last case.

Similarly prime divisors are points of $X$ and divisors are freely generated by prime divisors. Principal divisors are the zeros/poles of a non-zero function in $K(X)$.

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Generalization in analytic geometry

Now let us consider arbitrary dimension. We quickly discover that we cannot treat points of prime divisors:

We need to maintain that principal divisor are defined as the zeros/poles of non-zero functions. But in dimension $>1$, the preimage of zero/infinity are not points anymore, not even always complex manifold. If we let $X$ be a complex analytic variety, then the preimage of zero/infinity is always a codimension 1 complex analytic subvariety.

To make sure every non-zero function in $K(X)$ has a finite number of zeros/poles, it suffices to set that all those functions are algebraic/ratio of polynomial functions.

It suffices to set that $X$ is algebraic, i.e. $X$ is the analytification of a complex algebraic variety.

It suffices to set that $X$ is projective (in dimension 1, compact = projective). By GAGA, all projective complex analytic varieties are algebraic.

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Generalization in scheme Now let us consider schemes. Let $X$ be a scheme.

To make sure the existence of the function field, we set $X$ to be integral.

Analogous to the case of complex analytic varieties, we let prime divisors to be the set of codimension 1 integral closed subschemes (like codimension 1 complex analytic subvariety).

In above case, we define principal divisor to be the associated zeros/poles. So we need to find a way to define the order of a non-zero function $f$ along a codimension 1 integral closed subscheme $W$ with generic point $x$.

It can be done by setting that $\mathcal{O}_{X,x}$ is a discrete valuation ring.

In general, we need $X$ to be integral, Noetherian and regular in codimension 1.

To dinstinct from the next kind of divisor, we call this kind of divisor as the Weil divisor, named after André Weil.

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Generalization in locally ringed space

There are too many conditions on the scheme to succesfully define Weil divisor on it. So we want another kind of divisor that apply to general scheme.

Recall that the goal of developing theorem of divisors is to gain information in the order $$\mathrm{\mathop{Div}}(X)\to \mathrm{\mathop{PDiv}}(X)\to K(X)\to X$$ Hence we want [the relation between $K(X)$ and $\mathrm{\mathop{PDiv}}(X)$] and [the relation between $\mathrm{\mathop{PDiv}}(X)$ and $\mathrm{\mathop{Div}}(X)$] to be as close as possible.

Using this idea, we can

1. Just define principal divisor first and let a general divisor be a locally principal divisor. So the relation between $\mathrm{\mathop{PDiv}}(X)$ and $\mathrm{\mathop{Div}}(X)$ is very close.

2. define principal divisor $\mathrm{\mathop{div}}(f)$ with $f\in \mathcal{O}_X(U)$

to be [the element $f$ itself]

or [the ideal sheaf $(f)\subset \mathcal{O}_U$]

or [the closed subscheme $V(f)$ defined by the ideal sheaf]

In this way, there is a very close relation between the space of principal ideals of $\mathcal{O}_X(X)$ and $\mathrm{\mathop{PDiv}}(X)$, actually they are the same thing. This definition is much more fundamental/direct than the Weil divisor.

In the case of Weil divisor, given a principal ideal $f$, we calculate some kind of order to construct an associated divisor and hope this divisor recovers some information about $f$ or $(f)$.

Now we simply use $(f)$ itself as a divisor, much better. Now this kind of divisor (i.e. locally principal divisor), is called the Cartier divisor, named after Pierre Cartier.

Comparision of Cartier/Weil divisor:

Cartier divisor gives up the extremely simple group structure of $\mathrm{\mathop{Div}}(X)$: freely generated by the prime divisors, in return, it gains the a tighter map $\mathcal{K}_X(X)^* \to \mathrm{\mathop{Div}}(X)$, where $\mathcal{K}_X$ is the sheaf of meromorphic functions on $X$, and a much broader category of space that the theorem of Cartier divisor applies.