Divisors

On a variety \(X\), consider a rational function \(f \in K(X)\). This function has zeros at some points and is undefined at others. In essence, rational functions actually take the form of rational expressions, so the points where this function is undefined are a kind of pole, and we can also consider the order of such a pole. To describe this information systematically, we introduce the notion of a divisor.

For this, on the variety \(X\), for each codimension 1 irreducible subvariety \(Y\), at its generic point \(\eta_Y\), the stalk \(\mathcal{O}_{X, \eta_Y}\) must be a discrete valuation ring. For this to be well-defined, it suffices that the local rings of \(X\) are normal domains ([Commutative Algebra] §Integral Extensions, ⁋Definition 3), or, more generally, satisfy the \(R_1\) condition of [Commutative Algebra] §Regular Local Rings, ⁋Theorem 10 (Serre). However, for a variety to be normal is not that demanding a condition, and if a variety is not normal, we can circumvent this problem by considering the normalization of [Commutative Algebra] §Regular Local Rings, ⁋Definition 9; thus, in contexts where divisors appear, we implicitly understand \(X\) to be 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 represents the order of a zero or pole along the corresponding closed subvariety.

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

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

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 \(D \ge 0\).

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

Definition 3 For a rational function \(f \in K(X)^\times\), we define its principal divisor \(\divisor(f)\) as follows:

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

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

Here, since \(X\) is Noetherian, \(v_Y(f)\neq 0\) for only finitely many \(Y\), and therefore 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 \(x\), \(Y\) is defined by a single regular function \(\pi\). This \(\pi\) is called a local equation near \(x\). Now let us expand the rational function \(f\) near \(x\) as \(f = \pi^{v_Y(f)} \cdot u\). Here, all information about the zeros and poles of \(f\) is contained in \(\pi^{v_Y(f)}\); therefore, \(u\) is a function having neither zeros nor poles near \(x\), and \(v_Y(f)\) is the zero/pole order along \(Y\).

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

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

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

Example 5 On \(\mathbb{A}^1\), consider the rational function

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

Since along \(a_i\) this function has zeros of order \(n_i\), and along \(b_j\) poles of order \(m_j\), the principal divisor of this function is

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

Here, \((a_i)\) is the divisor representing the point \(a_i\); positive coefficients indicate zeros and negative coefficients indicate poles.

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

Proof

For \(f, g \in K(X)^\times\) 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 from \(\Div(X)\) properties of \(X\). However, for this purpose, \(\Div(X)\) is unnecessarily large. In any case, since we understand elements of \(K(X)^\times\) reasonably well, we will treat a divisor obtained by operating on a specific divisor with a divisor obtained from an element of \(K(X)^\times\) as the same as the original divisor. That is, we define the following.

Definition 7 Two Weil divisors \(D_1, D_2\) are linearly equivalent if \(D_1 - D_2 = \divisor(f)\) for some rational function \(f \in K(X)^\times\). 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 \(D_1 - D_2 = \divisor(f)\) for some \(f\), 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, \(D_1 - D_2 = \divisor(f)\) and \(D_2 - D_3 = \divisor(g)\) for some \(f, g\). Then \(D_1 - D_3 = \divisor(fg)\), so \(D_1 \sim D_3\).

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

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

To see which elements of \(\Div(X)\) coincide in \(\Cl(X)\), it suffices to examine which divisors are linearly equivalent to \(0\). This is essentially [Algebraic Topology] §Homotopy, ⁋Definition 2 translated to algebraic geometry. 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) \(H(x, 0) = f_0(x)\) and (ii) \(H(x, 1) = f_1(x)\). That is, while \(t\) moves from \(0\) to \(1\), the family of functions \((H(-,t))_t\) continuously deforms \(f_0\) into \(f_1\), and taking the existence of such a family connecting two objects as an equivalence relation is the idea of homotopy. To translate this concept into algebraic geometry, instead of \([0,1]\) we use \(\mathbb{P}^1\) as the parameter space. More concretely, viewing an element of \(K(X)^\times\), \(f\), as a rational map \(X \dashrightarrow \mathbb{P}^1\), we can consider the closure of the graph over the open set where \(f\) is defined, \(U\subseteq X\):

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

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

That is, looking at this motivation, one can say that while \(\Cl(X)\) is smaller than \(\Div(X)\), it 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 this is the principal divisor of some function. Now, since \(Y_i\) 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 necessarily principal. That is, \(I(Y_i)=(f_i)\) for some function \(f_i\in \mathbb{K}[\x_1,\ldots, \x_n]\), and from this each \(Y_i\) is \(Z(f_i)\). Here, 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, let us take an arbitrary hyperplane class, say \(H=Z(\x_0)\), as a reference. We will first show that, for an arbitrary degree \(d\) homogeneous polynomial \(F\), the hypersurface \(Z(F)\) it defines is linearly equivalent to \(dH\). To see this, considering the function \(F/\x_0^d\in K(\mathbb{P}^n)^\times\), we have

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

Therefore, a Weil divisor \(D=\sum n_i Y_i\) is determined by the orders \(n_i\) and, for each \(Y_i\), the degree \(d_i\) of the homogeneous polynomial defining it. Taking this as intuition, let us 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 \(Y_i\). Here, since every hypersurface is the zero set of some homogeneous polynomial, there is no problem in defining it this way. Moreover, since we know by §Rational Maps, ⁋Example 4 that any rational function on \(\mathbb{P}^n\) is expressed as a ratio of homogeneous polynomials \(F,G\) of the same degree, \(F/G\), it necessarily holds for a principal divisor \(\divisor(F/G)\) that \(\deg(\divisor(F/G))=0\), so this definition is well-defined.

Our claim is that \(\deg\) is an isomorphism. First, that \(\deg\) is surjective is clear since the images of the \(dH\) go to \(d\). For injectivity, we must show that if \(\deg(D)=0\), then \(D\) is a principal divisor. However, we have seen earlier that \(D \sim dH\), and thus for this to map under \(\deg\) to \(0\), we must have \(d=0\). That is, \(D\) is linearly equivalent to \(0\).

Intuitively, the reason this differs from \(\mathbb{A}^n\) is that global regular functions defined on \(\mathbb{P}^n\) are constant functions. That is, if a function defined on \(\mathbb{P}^n\) has a zero at a certain point, this function must also have a pole at another point, and the sum of the orders of the zeros and the sum of the orders of the poles must be equal. That \(\Cl(\mathbb{P}^n)\cong \mathbb{Z}\) means that, for \(d\neq 0\), the divisors \(dH\) are essentially all the non-principal divisors on \(\mathbb{P}^n\), while the remaining divisors \(\sum n_i D_i\) merely distribute this order \(d\) via a rational function \(f\) to each \(D_i\).

Cartier Divisors

Weil divisors are geometrically intuitive, but they have the problem that they do not work well on singular varieties. For example, on the cone \(X=Z(\x^2 + \y^2 - \z^2) \subseteq \mathbb{A}^3\), let us imagine defining a principal divisor following the explanation after Definition 3. At a smooth point of this cone, say \((1,0,1)\), if we consider a codimension \(1\) subvariety, say \(X\cap Z(\y)\), this subvariety is defined near this point by the local equation \(\y=0\), and therefore, given an arbitrary rational function \(f\), we can extract information about its zeros and poles with respect to \(\y\). The problem arises at a singular point. For instance, consider, passing through the origin \((0,0,0)\), the codimension \(1\) subvariety \(L=(t,0,t)\). To represent this, both equations \(\y=0\) and \(\x-\z=0\) are required. Indeed, on \(X\), since \((\x-\z)(\x+\z)=-\y^2\) holds, we have \(\divisor(\x-\z)=2L\), and the ideal of \(L\), \((\y,\x-\z)\), is not principal in \(\mathcal{O}_{X,(0,0,0)}\), so no single equation defining \(L\) near the origin exists.

That is, the core problem is that on a singular variety, even for a codimension \(1\) part, cases arise where it cannot be expressed locally by a single equation. Therefore, we simply restrict our attention to objects that are locally expressed by a single equation. To do this, as always, we define them via a suitable kind of gluing.

Definition 12 A Cartier divisor on a variety \(X\) consists of the following data:

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

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

Here, for two data \(\{(U_i, f_i)\}\) and \(\{(V_j, g_j)\}\), 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, these two data are regarded as representing the same Cartier divisor.

By definition, a Cartier divisor is a collection of only those divisors given locally by a single equation, that is, locally principal divisors. The data \(U_i\) explicitly specifies precisely which open set it is local to, and even by which single function it is defined on that open set. According to this, the line \(L\) examined earlier cannot be represented by a single equation on an open neighborhood of the origin, and thus is not a Cartier divisor.

Example 13 In the cone example above, if we consider the line \(L'=(t,0,-t)\), this is not a Cartier divisor for the same reason as \(L\). However, the 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 divisor with the added locally principal condition. Concretely, if a Cartier divisor \(\{(U_i, f_i)\}\) is given, on each \(U_i\) we can consider the principal Weil divisor of \(f_i\), \(\divisor(f_i)\). Here, since on \(U_i \cap U_j\), \(f_i/f_j\) is invertible, for every subvariety meeting \(U_i \cap U_j\) of codimension \(1\), \(Y\), we have \(v_Y(f_i)=v_Y(f_j)\), and therefore we can glue these to define a Weil divisor.

We saw above that the converse does not hold in general, but at the same time we can see that in the smooth case, since the stalk will be a regular local ring, there is no particular problem. That is, 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 of divisors and sending principal Cartier divisors to principal Weil divisors.

Essentially, Weil divisors play a role like homology classes, namely particular subsets of the space \(X\), while Cartier divisors, by their definition, play a role like cohomology classes, namely functions defined (locally) on the space \(X\). From this perspective, this proposition can be thought of as a kind of Poincaré duality. ([Algebraic Topology] §Poincaré Duality, ⁋Theorem 11)

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

Definition 15 For a rational function \(f \in K(X)^\times\), 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 Cartier divisor.

Let us 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.