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 genuinely rational expressions, the points where \(f\) is undefined are poles, and we may speak of the order of each pole. To encode this information systematically, we introduce the notion of a divisor.

For this to work, for each 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. This is well-defined provided the local rings of \(X\) are normal domains ([Commutative Algebra] §Integral Extensions, ⁋Definition 3), or more generally that they satisfy the \(R_1\) condition of [Commutative Algebra] §Regular Local Rings, ⁋Theorem 10. Normality is not a very stringent condition, and if a variety fails to be normal we may circumvent the issue by passing to the normalization of [Commutative Algebra] §Regular Local Rings, ⁋Definition 9; hence in any context where divisors appear we implicitly assume \(X\) 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 coefficient of each term records 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 denote this by \(D \ge 0\).

As mentioned above, the basic idea of a Weil divisor is to encode the zeros and poles of a rational function \(f \in \mathbb{K}(X)^\times\). Such a divisor is called a principal divisor.

Definition 3 The principal divisor \(\divisor(f)\) of a rational function \(f \in \mathbb{K}(X)^\times\) is defined by

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

where the sum ranges over all codimension 1 irreducible closed subvarieties \(Y\) of \(X\), and \(v_Y(f)\) is the integer giving the order of the zero or pole of \(f\) along \(Y\).

Since \(X\) is Noetherian, only finitely many \(Y\) satisfy \(v_Y(f)\neq 0\), so the above sum is finite and defines a Weil divisor as in Definition 1.

If \(Y\) is a codimension 1 closed subvariety and \(X\) is smooth at a point \(x \in Y\), then near \(x\) the subvariety \(Y\) is defined by a single regular function \(\pi\). This \(\pi\) is called a local equation near \(x\). Expanding the rational function \(f\) near \(x\) as \(f = \pi^{v_Y(f)} \cdot u\), we see that all information about the zeros and poles of \(f\) is carried by \(\pi^{v_Y(f)}\), while \(u\) is a function with neither zeros nor poles near \(x\); thus \(v_Y(f)\) is the zero/pole order along \(Y\).

To make this 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 ring. Then \(v_Y(f)\) is exactly the valuation of \(f\) in 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 why we require \(X\) to be a normal variety.

Example 4 Consider the regular function \(f(\x, \y) = \x^2 \y\) 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 \(D_2\) it has order 1, so the principal divisor associated 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 of the point \(a_i\); positive coefficients indicate zeros and negative coefficients indicate poles.

Proposition 6 \(\divisor: \mathbb{K}(X)^\times \rightarrow \Div(X)\) is a group homomorphism.

Proof

For \(f, g \in \mathbb{K}(X)^\times\) and each \(Y\),

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

so

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

Our goal is to extract properties of \(X\) from \(\Div(X)\). However, \(\Div(X)\) is unnecessarily large for this purpose. Since we already understand the elements of \(\mathbb{K}(X)^\times\) reasonably well, we shall regard a divisor obtained from another by adding a principal divisor as equivalent to the original one. 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 \(f \in \mathbb{K}(X)^\times\) such that \(D_1 - D_2 = \divisor(f)\). We denote this by \(D_1 \sim D_2\).

Then the following holds.

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

Proof

First, \(D - D = 0 = \divisor(1)\), so \(D \sim D\). 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, assume \(D_1 \sim D_2\) and \(D_2 \sim D_3\). That is, there exist \(f, g\) with \(D_1 - D_2 = \divisor(f)\) and \(D_2 - D_3 = \divisor(g)\). Then \(D_1 - D_3 = \divisor(fg)\), so \(D_1 \sim D_3\).

Definition 9 We define the divisor class group \(\Cl(X)\) of \(X\) as the quotient of \(\Div(X)\) by linear equivalence:

\[\Cl(X) = \Div(X) / \{\divisor(f) \mid f \in \mathbb{K}(X)^\times\}\]

To determine which elements of \(\Div(X)\) coincide in \(\Cl(X)\), it suffices to examine which divisors are linearly equivalent to \(0\). This is essentially the algebro-geometric translation of [Algebraic Topology] §Homotopy, ⁋Definition 2. Two continuous functions \(f_0, f_1:X \rightarrow Y\) are homotopic if there exists a continuous function

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

satisfying (i) \(H(x, 0) = f_0(x)\), (ii) \(H(x, 1) = f_1(x)\). That is, as \(t\) moves from \(0\) to \(1\), the family of functions \((H(-,t))_t\) continuously deforms \(f_0\) into \(f_1\), and homotopy takes the existence of such a family connecting two objects as an equivalence relation. To translate this notion into algebraic geometry, we use \(\mathbb{P}^1\) as the parameter space in place of \([0,1]\). More concretely, viewing an element \(f \in \mathbb{K}(X)^\times\) as a rational map \(X \dashrightarrow \mathbb{P}^1\), we may consider the closure of its graph over the open set \(U\subseteq X\) where \(f\) is defined:

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

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

From this motivation, we may 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, let an arbitrary Weil divisor \(D=\sum n_i Y_i\) be given. We must show that it 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 \(\codim Y_i=\codim I(Y_i)=1\). But \(\mathbb{K}[\x_1,\ldots, \x_n]\) is a unique factorization domain by [Ring Theory] §Polynomial Rings, ⁋Theorem 16, so every height 1 prime ideal is principal. That is, there exists \(f_i\in \mathbb{K}[\x_1,\ldots, \x_n]\) such that \(I(Y_i)=(f_i)\), and hence each \(Y_i\) is \(Z(f_i)\). Since \(f_i\) is a prime element, \(\divisor(f_i)=Y_i\), and therefore \(D=\divisor(\prod_i f_i^{n_i})\) is a principal divisor.

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

Example 11 \(\Cl(\mathbb{P}^n) \cong \mathbb{Z}\). To verify this, fix a hyperplane class, say \(H=Z(\x_0)\), as a reference. We first show that a hypersurface \(Z(F)\) defined by an arbitrary homogeneous polynomial \(F\) of degree \(d\) is linearly equivalent to \(dH\). Indeed, consider the function \(F/\x_0^d\in \mathbb{K}(\mathbb{P}^n)^\times\); then

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

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 the \(Y_i\). Motivated by this, define the map

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

where \(\deg(Y_i)\) is the degree of the homogeneous polynomial defining \(Y_i\). Since every hypersurface is the zero set of some homogeneous polynomial, this definition is unproblematic. Moreover, by [Rational Maps] §Rational Maps, ⁋Example 4, any rational function on \(\mathbb{P}^n\) is a ratio \(F/G\) of homogeneous polynomials of the same degree; hence for every principal divisor \(\divisor(F)\) we have \(\deg(\divisor(F))=0\), and the map is well-defined.

We claim that \(\deg\) is an isomorphism. Surjectivity is immediate, since the image of \(dH\) is \(d\). 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 \(d=0\). Hence \(D\) is linearly equivalent to \(0\).

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

Cartier Divisors

Weil divisors are geometrically intuitive, but they do not behave well on singular varieties. For example, on the cone \(X=Z(\x^2 + \y^2 - \z^2) \subseteq \mathbb{A}^3\), imagine defining a principal divisor following the explanation after Definition 3. At a smooth point of this cone, say \((1,0,1)\), a codimension 1 subvariety such as \(X\cap Z(\y)\) is defined by the local equation \(\y=0\) near this point, so for any rational function \(f\) we can extract zero and pole information with respect to \(\y\). The problem arises at the singular point. Consider the codimension 1 subvariety \(L=(t,0,t)\) passing through the origin \((0,0,0)\). To represent \(L\) requires both equations \(\y=0\) and \(\x-\z=0\). Indeed, since \((\x-\z)(\x+\z)=-\y^2\) holds on \(X\), we have \(\divisor(\x-\z)=2L\), and the ideal \((\y,\x-\z)\) of \(L\) is not principal in \(\mathcal{O}_{X,(0,0,0)}\); hence no single equation defines \(L\) near the origin.

The core problem is that on a singular variety, a codimension 1 subvariety need not be expressible locally by a single equation. Therefore we simply restrict our attention to those objects that are locally principal. As usual, we define this by means of an appropriate gluing.

Definition 12 A Cartier divisor on a variety \(X\) is given by the following data:

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

where \(\{U_i\}\) is an open cover of \(X\), each \(f_i \in \mathbb{K}(X)^\times\) 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 common refinement \(\{W_k\}\) on which each \(f_i/g_j\) is regular and non-vanishing.

By definition, a Cartier divisor collects only those divisors given locally by a single equation, i.e., locally principal divisors. The data explicitly specify which open set we localize to and which single function defines the divisor there. According to this, the line \(L\) examined above is not a Cartier divisor, since it cannot be represented by a single equation on any open neighborhood of the origin.

Example 13 In the cone example above, the line \(L'=(t,0,-t)\) 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 by the zero set of \(\y\).

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

We saw above that the converse does not hold in general, but we also see that there is no difficulty in the smooth case, since the stalk is then a regular local ring. Hence the following holds.

Proposition 14 On a smooth variety \(X\), Weil divisors and Cartier divisors are in natural bijection; this correspondence is a group isomorphism preserving addition and sending principal Cartier divisors to principal Weil divisors.

Essentially, a Weil divisor plays a role analogous to a homology class, being a certain subset of the space \(X\), while a Cartier divisor, by its very definition, consists of functions defined locally on \(X\), playing a role analogous to a cohomology class. From this perspective, the proposition may be regarded as a kind of Poincaré duality. ([Algebraic Topology] §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 \(f \in \mathbb{K}(X)^\times\), the principal Cartier divisor \(\divisor(f)\) is defined as \(\{(X, f)\}\).

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

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

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

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