Divisors

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Consider a rational function \(f \in \mathbb{K}(X)\) defined on a variety \(X\). This function has zeros at some points and is undefined at others. Since rational functions are essentially rational expressions, the points where this function is undefined are poles, and we can also speak of the order of each such pole. To describe this information systematically, we introduce the notion of a divisor.

For this to work, for every codimension 1 irreducible subvariety \(Y\) of \(X\), the stalk \(\mathcal{O}_{X, \eta_Y}\) at its generic point \(\eta_Y\) must be a discrete valuation ring. For this to be well-defined, it suffices that \(X\) is a normal domain (§Integral Extensions, ⁋Definition 3), or more generally that the \(R_1\) condition of §Regular Local Rings, ⁋Theorem 11 (Serre) is satisfied. However, normality is not such a stringent condition for a variety; if a variety is not normal, we can circumvent this issue by considering the normalization of §Regular Local Rings, ⁋Definition 10. Thus, in any context where divisors appear, we implicitly assume that \(X\) is normal.

Weil Divisors

We first examine the most intuitive definition of a divisor, the Weil divisor. This is a formal sum of codimension 1 closed subvarieties with integer coefficients, where the integer coefficient of each term indicates the order of a zero or pole along the corresponding closed subvariety.

Definition 1 A Weil divisor on a variety \(X\) is a formal \(\mathbb{Z}\)-linear combination of codimension 1 (irreducible) closed subvarieties of \(X\):

\[D = \sum_{i=1}^{r} n_i Y_i.\]

We denote the set of Weil divisors by \(\Div(X)\).

Then \(\Div(X)\) forms an abelian group under addition.

Definition 2 A Weil divisor \(D = \sum_i n_i Y_i\) is effective if all \(n_i \ge 0\). We write this as \(D \ge 0\).

Meanwhile, as mentioned above, the basic idea of a Weil divisor is to encode information about the zeros and poles of a rational function \(f \in \mathbb{K}(X)^\ast\). This is called a principal divisor.

Definition 3 We define the principal divisor \(\divisor(f)\) of a rational function \(f \in \mathbb{K}(X)^\ast\) as follows:

\[\divisor(f) = \sum_{Y} v_Y(f) \cdot Y\]

Here the sum is over all codimension 1 irreducible closed subvarieties \(Y\) of \(X\), and \(v_Y(f)\) is the integer representing the order of the zero or pole that \(f\) has along \(Y\).

If \(Y\) is a smooth subvariety of codimension 1, then in a neighborhood of each point \(x\) of \(Y\), the subvariety \(Y\) is defined by a single regular function \(\pi\). We call this \(\pi\) a local equation in a neighborhood of \(P\). Now expand the rational function \(f\) in a neighborhood of \(x\) as \(f = \pi^{v_Y(f)} \cdot u\). Here all information about the zeros and poles of \(f\) is contained in \(\pi^{v_Y(f)}\), so \(u\) is a function having neither zeros nor poles in a neighborhood of \(x\), and \(v_Y(f)\) is the zero/pole order along \(Y\).

To make this mathematically rigorous, the stalk \(\mathcal{O}_{X, \eta}\) of the structure sheaf at the generic point \(\eta\) of \(Y\) is a discrete valuation ring, and the local equation \(\pi\) corresponds to a uniformizer of this discrete valuation ring. Then \(v_Y(f)\) is exactly the valuation of this discrete valuation ring. For this to make sense, the stalk \(\mathcal{O}_{X, \eta}\) at the generic point of \(Y\) must be a discrete valuation ring, and this is precisely the reason we assume \(X\) to be a normal variety.

Example 4 Consider the regular function \(f(\x, \y) = \x^2 \y\) defined on \(\mathbb{A}^2\). This function has no poles, and its zeros lie only along the two irreducible closed subvarieties \(D_1=Z(\x)\) and \(D_2=Z(\y)\). Along \(D_1\) the zero has order \(2\), and along \(D_2\) the zero has order \(1\), so the principal divisor corresponding to \(f\) is

\[\divisor(f)=2D_1+D_2\]

.

::: Example 5 Consider the rational function on \(\mathbb{A}^1\)

\[g(\x) = \frac{(\x-a_1)^{n_1} \cdots (\x-a_k)^{n_k}}{(\x-b_1)^{m_1} \cdots (\x-b_l)^{m_l}}.\]

This function has zeros of order \(n_i\) at the \(a_i\) and poles of order \(m_j\) at the \(b_j\), so its principal divisor is

\[\divisor(g) = n_1(a_1) + \cdots + n_k(a_k) - m_1(b_1) - \cdots - m_l(b_l).\]

Here \((a_i)\) denotes the divisor corresponding to the point \(a_i\); positive coefficients indicate zeros and negative coefficients indicate poles.

Proposition 6 \(\divisor: \mathbb{K}(X)^\ast \to \operatorname{Div}(X)\) is a group homomorphism.

Proof

For \(f, g \in \mathbb{K}(X)^\ast\), for each \(Y\) we have

\[v_Y(fg) = v_Y(f) + v_Y(g)\]

and therefore

\[\divisor(fg) = \sum_Y v_Y(fg) \cdot Y = \sum_Y (v_Y(f) + v_Y(g)) \cdot Y = \divisor(f) + \divisor(g)\]

as desired.

Our goal is to extract properties of \(X\) from \(\Div(X)\). However, for this purpose \(\Div(X)\) is unnecessarily large. Since we already understand the elements of \(\mathbb{K}(X)^\ast\) reasonably well, we regard a divisor as equivalent to the divisor obtained from it by adding or subtracting divisors coming from elements of \(\mathbb{K}(X)^\ast\). That is, we make the following definition.

Definition 7 Two Weil divisors \(D_1, D_2\) are linearly equivalent if there exists a rational function \(f \in \mathbb{K}(X)^\ast\) such that \(D_1 - D_2 = \divisor(f)\). We write this as \(D_1 \sim D_2\).

Then the following holds.

Proposition 8 Linear equivalence \(\sim\) is an equivalence relation on \(\operatorname{Div}(X)\).

Proof

First, since \(D - D = 0 = \divisor(1)\), we have \(D \sim D\). Also, if \(D_1 \sim D_2\), then there exists \(f\) with \(D_1 - D_2 = \divisor(f)\), and for this \(f\) we have \(D_2 - D_1 = \divisor(f^{-1})\). Finally, suppose \(D_1 \sim D_2\) and \(D_2 \sim D_3\). That is, there exist \(f, g\) such that \(D_1 - D_2 = \divisor(f)\) and \(D_2 - D_3 = \divisor(g)\). Then \(D_1 - D_3 = \divisor(fg)\), so \(D_1 \sim D_3\).

Definition 9 The divisor class group \(\Cl(X)\) of \(X\) is defined as the quotient of \(\operatorname{Div}(X)\) by linear equivalence:

\[\Cl(X) = \operatorname{Div}(X) / \{\divisor(f) : f \in \mathbb{K}(X)^\ast\}\]

To determine which elements of \(\Div(X)\) become equal in \(\Cl(X)\), it suffices to examine which divisors are linearly equivalent to \(0\). This is essentially the algebraic-geometric translation of §Homotopy, ⁋Definition 9. For a fixed space \(X\) and a subspace \(A\), we say that \(A\) is a deformation retract of \(X\) if there exists a continuous function

\[H:[0,1]\times X\rightarrow X\]

satisfying (i) \(H(0, x) = x\), (ii) \(H(1, x) \in A\), and (iii) \(H(t, a) = a\) whenever \(a \in A\). In other words, at \(t=0\) we have all of \(X\), at \(t=1\) we have \(A\), and \(A\) itself remains fixed during the deformation. To translate this notion into algebraic geometry, we use \(\mathbb{P}^1\) as the parameter space in place of \([0,1]\). More specifically, viewing an element \(f \in \mathbb{K}(X)^\ast\) as a rational map \(X \dashrightarrow \mathbb{P}^1\), we may consider its graph

\[\Gamma_f =\{(x, t)\in X\times \mathbb{P}^1\mid f(x)=t\}\subset X \times \mathbb{P}^1\]

Then, pulling back the coordinate of \(\mathbb{P}^1\) via the canonical projection \(\pr_2:\Gamma_f\rightarrow \mathbb{P}^1\) gives \(f\) itself, and thus \(\divisor(f)\) measures, when viewed on \(\Gamma_f\), the difference between the \(t=0\) section and the \(t=\infty\) section.

From this motivation, we can say that \(\Cl(X)\) is smaller than \(\Div(X)\), yet still retains information about the properties of \(X\).

Example 10 \(\Cl(\mathbb{A}^n) = 0\). To verify this, suppose an arbitrary Weil divisor \(D=\sum n_i Y_i\) is given. We need to show that this is the principal divisor of some function. Since each \(Y_i\) is an irreducible closed subvariety, it corresponds to a prime ideal \(I(Y_i)\) of the coordinate ring \(\mathbb{K}[\x_1,\ldots, \x_n]\), and \(\codim Y_i=\codim I(Y_i)=1\). Now \(\mathbb{K}[\x_1,\ldots, \x_n]\) is a unique factorization domain by §Polynomial Rings, ⁋Theorem 16, so every height 1 prime ideal is necessarily principal. That is, there exists a function \(f_i\in \mathbb{K}[\x_1,\ldots, \x_n]\) such that \(I(Y_i)=(f_i)\), and hence each \(Y_i\) is \(Z(f_i)\).

Of course, to understand what kind of information \(\Cl(X)\) carries, we should consider the case where \(\Cl(X)\neq 0\).

Example 11 \(\Cl(\mathbb{P}^n) \cong \mathbb{Z}\). To verify this, let us take an arbitrary hyperplane class, say \(H=Z(\x_0)\), as our reference. We first show that the hyperplane \(Z(F)\) defined by an arbitrary homogeneous polynomial \(F\) of degree \(d\) is equal to \(dH\). To see this, consider the function \(F/\x_0^d\in \mathbb{K}(\mathbb{P}^n)^\ast\); then

\[\divisor(F/\x_0^d)=\divisor(F)-d\cdot \divisor(\x_0)=Z(F)-dH\]

holds.

Thus a Weil divisor \(D=\sum n_i Y_i\) is determined by the orders \(n_i\) and the degrees \(d_i\) of the homogeneous polynomials defining \(Y_i\). Taking this as our intuition, we define the map

\[\deg: \Cl(\mathbb{P}^n) \rightarrow \mathbb{Z};\qquad D=\sum n_i Y_i \mapsto \sum n_i \deg(Y_i)\]

Here \(\deg(Y_i)\) is the degree of the homogeneous polynomial defining \(Y_i\). Since every hypersurface is the zero set of some homogeneous polynomial, there is no ambiguity in this definition. Moreover, we know from §Rational Maps, ⁋Example 4 that any rational function on \(\mathbb{P}^n\) can be written as a ratio \(F/G\) of homogeneous polynomials \(F,G\) of the same degree, so for any principal divisor \(\divisor(F)\) we necessarily have \(\deg(\divisor(F))=0\); hence this definition is well-defined.

Our claim is that \(\deg\) is an isomorphism. First, \(\deg\) is surjective because the images of the \(dH\) are simply \(d\), which is obvious. For injectivity, we must show that any \(D\) with \(\deg(D)=0\) is a principal divisor. But we have already seen that \(D \sim dH\), and for this to map to \(0\) under \(\deg\) we must have \(d=0\). That is, \(D\) is linearly equivalent to \(0\).

Intuitively, the reason this differs from \(\mathbb{A}^n\) is that every global regular function on \(\mathbb{P}^n\) is constant. In other words, if a function defined on \(\mathbb{P}^n\) has a zero at some point, it must also have a pole at some other point, and the sum of the orders of the zeros equals the sum of the orders of the poles. The fact that \(\Cl(\mathbb{P}^n)\cong \mathbb{Z}\) means that the \(dH\) with \(d\neq 0\) are essentially all the non-principal divisors of \(\mathbb{P}^n\), while the remaining divisors \(\sum n_i D_i\) merely distribute this order \(d\) among the various \(D_i\) via a rational function \(f\).

Cartier Divisors

Weil divisors are geometrically intuitive, but they do not work well on singular varieties. For example, consider the cone \(X=Z(\x^2 + \y^2 - \z^2) \subset \mathbb{A}^3\), and imagine defining a principal divisor following the explanation after Definition 3. At a smooth point of this cone, say \((1,0,1)\), consider a codimension \(1\) subvariety such as \(X\cap Z(\y)\); this subvariety is defined locally near this point by the equation \(\y=0\), so for any rational function \(f\) we can extract the zero and pole information with respect to \(\y\) here. The problem arises at singular points. For instance, near the origin \((0,0,0)\), consider the codimension \(1\) subvariety \(L=(t,0,t)\). To represent this, both equations \(\y=0\) and \(\x-\z=0\) are needed, which means we cannot define \(v_L(f)\) properly. More precisely, \(\mathcal{O}_{X,(0,0,0)}\) is not a (dimension \(1\)) regular local ring, and hence not a discrete valuation ring, so we cannot define a valuation in the first place.

That is, the key issue is that on a singular variety, a codimension \(1\) subvariety may fail to be expressed locally by a single equation. Therefore, we simply restrict our attention to those objects that are locally given by a single equation. As always, we define this by an appropriate kind of gluing.

Definition 12 A Cartier divisor on a variety \(X\) is the data

\[\{(U_i, f_i)\}_{i \in I}\]

consisting of an open cover \(\{U_i\}\) of \(X\), where each \(f_i \in \mathbb{K}(X)^\ast\) is a nonzero rational function, and for all \(i, j\), the ratio \(f_i/f_j\) is regular and non-vanishing (hence invertible) on \(U_i \cap U_j\).

Two such data \(\{(U_i, f_i)\}\) and \(\{(V_j, g_j)\}\) represent the same Cartier divisor if there exists a refinement \(\{W_k\}\) of the two open covers such that \(f_i/g_j\) is regular and non-vanishing on it.

By definition, a Cartier divisor collects only those codimension 1 subvarieties that are locally principal. The data \(U_i\) explicitly specifies which open sets we localize to, and which single function defines the divisor on each such open set. Consequently, the line \(L\) we examined earlier is not a Cartier divisor, because it cannot be expressed as a single equation in any open neighborhood of the origin.

Example 13 In the example of the cone above, consider the line \(L'=(t,0,-t)\); this is not a Cartier divisor for the same reason as \(L\). However, their sum \(L+L'\) is a Cartier divisor, because \(L+L'\) is defined as the zero set of \(\y\).

As mentioned above, a Cartier divisor can be thought of as a codimension 1 subvariety with the additional locally principal condition. Specifically, if a Cartier divisor \(\{(U_i, f_i)\}\) is given, we can consider the principal Weil divisor \(\divisor(f_i)\) of \(f_i\) on each \(U_i\). Since \(f_i/f_j\) is invertible on \(U_i \cap U_j\), we have \(v_Y(f_i)=v_Y(f_j)\) for every codimension \(1\) subvariety \(Y\), and thus we can glue these together to define a Weil divisor.

We have seen above that the converse does not hold in general, but at the same time we can see that there is no problem when the variety is smooth, because the stalk becomes a regular local ring. That is, the following holds.

Proposition 14 On a smooth variety \(X\), Weil divisors and Cartier divisors are in natural one-to-one correspondence.

Essentially, a Weil divisor plays a role similar to a homology class, as it consists of certain subsets of the space \(X\), whereas a Cartier divisor plays a role similar to a cohomology class, since by definition it is given by functions defined (locally) on the space \(X\). From this perspective, this proposition may be thought of as a kind of Poincaré duality. (§Poincaré Duality, ⁋Theorem 11)

We now define linear equivalence and the divisor class group for Cartier divisors, just as we did for Weil divisors. To do so, we first need the following.

Definition 15 For a rational function \(f \in \mathbb{K}(X)^\ast\), the principal Cartier divisor \(\divisor(f)\) is defined as \(\{(X, f)\}\).

Then the following definition is the Cartier version of Definition 7.

Definition 16 Two Cartier divisors \(D_1, D_2\) are linearly equivalent if \(D_1 - D_2\) is a principal divisor.

We denote the group of Cartier divisors by \(\CaDiv(X)\), and the subgroup generated by principal divisors by \(\Prin(X)\). Then the Cartier divisor class group is

\[\CaCl(X) = \CaDiv(X) / \Prin(X)\]

By Proposition 14, on a smooth variety we have \(\CaCl(X) \cong \Cl(X)\).

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References

[Har] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Sha] I. R. Shafarevich, Basic Algebraic Geometry I: Varieties in Projective Space, Springer, 2013.