This post was machine-translated from Korean by Kimi CLI. The translation may contain errors. 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 ratios of regular functions, the points where it is undefined are a kind of pole, and we can also consider the order of such poles. To describe this information systematically, we introduce the notion of a divisor.
For this to work, for each codimension 1 irreducible subvariety $Y$ of a variety $X$, 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 ([Commutative Algebra] §Integral Extensions, ⁋Definition 3), or more generally that the $R_1$ condition of [Commutative Algebra] §Regular Local Rings, ⁋Theorem 10 is satisfied. However, normality of a variety is not such a stringent condition, and if a variety is not normal we can avoid this issue by passing to its normalization ([Commutative Algebra] §Regular Local Rings, ⁋Definition 9); therefore, in any context 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 indicates the order of the 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 the zeros and poles of a rational function \(f \in \mathbb{K}(X)^\ast\). This is called a principal divisor.
Definition 3 The principal divisor \(\divisor(f)\) of a rational function \(f \in \mathbb{K}(X)^\ast\) is defined by
\[\divisor(f) = \sum_{Y} v_Y(f) \cdot Y\]where the sum runs over all codimension 1 irreducible closed subvarieties \(Y\) of \(X\), and \(v_Y(f)\) is the integer indicating 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 \(P\). Expanding the rational function \(f\) in a neighborhood of \(x\) as \(f = \pi^{v_Y(f)} \cdot u\), all information about the zeros and poles of \(f\) is contained in \(\pi^{v_Y(f)}\); thus \(u\) 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 precisely the valuation 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 exactly the reason we take \(X\) 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; therefore 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 \(n_i\) along the \(a_i\) and poles of order \(m_j\) along 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
\[\divisor(fg) = \sum_Y v_Y(fg) \cdot Y = \sum_Y (v_Y(f) + v_Y(g)) \cdot Y = \divisor(f) + \divisor(g)\]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 \(\mathbb{K}(X)^\ast\) to a reasonable degree, we will regard a divisor as equivalent to the one obtained from it by adding a divisor coming from an element 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, \(D - D = 0 = \divisor(1)\), 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 \(f, g\) 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 know 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 analogue of [Algebraic Topology] §Homotopy, ⁋Definition 9. For a fixed space \(X\) and a subspace \(A\), \(A\) is a deformation retract of \(X\) if there exists a continuous map
\[H:[0,1]\times X\rightarrow X\]satisfying (i) \(H(0, x) = x\), (ii) \(H(1, x) \in A\), (iii) if \(a \in A\) then \(H(t, a) = a\). In other words, at \(t=0\) we have all of \(X\), at \(t=1\) we obtain \(A\), 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 concretely, 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 \(\pr_2:\Gamma_f\rightarrow \mathbb{P}^1\) yields \(f\) itself; consequently, \(\divisor(f)\) measures, on \(\Gamma_f\), the difference between the \(t=0\) section and the \(t=\infty\) section.
Thus, 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 this is the principal divisor of some function. Since each \(Y_i\) is an irreducible closed subvariety, it corresponds to a prime ideal \(I(Y_i)\) in 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 [Ring Theory] §Polynomial Rings, ⁋Theorem 16; hence every height 1 prime ideal is necessarily principal. That is, there exists a function \(f \in \mathbb{K}[\x_1,\ldots, \x_n]\) such that \(I(Y_i)=(f_i)\), and consequently each \(Y_i\) is \(Z(f_i)\).
Of course, to understand what kind of information \(\Cl(X)\) carries, we should consider the case \(\Cl(X)\neq 0\).
Example 11 \(\Cl(\mathbb{P}^n) \cong \mathbb{Z}\). To verify this, let us fix a hyperplane class, for instance \(H=Z(\x_0)\). We first show that the hyperplane \(Z(F)\) defined by an arbitrary homogeneous polynomial \(F\) of degree \(d\) equals \(dH\). Consider the function \(F/\x_0^d\in \mathbb{K}(\mathbb{P}^n)^\ast\); then
\[\divisor(F/\x_0^d)=\divisor(F)-d\cdot \divisor(\x_0)=Z(F)-dH\]Hence 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 \(Y_i\). Guided by this intuition, define
\[\deg: \Cl(\mathbb{P}^n) \rightarrow \mathbb{Z};\qquad D=\sum n_i Y_i \mapsto \sum n_i \deg(Y_i)\]where \(\deg(Y_i)\) is the degree of the homogeneous polynomial defining \(Y_i\). Every hypersurface is the zero set of some homogeneous polynomial, so there is no ambiguity in this definition. Moreover, by §Rational Maps, ⁋Example 4 we know that any rational function on \(\mathbb{P}^n\) is expressed as a ratio \(F/G\) of homogeneous polynomials \(F,G\) of the same degree; therefore for a principal divisor \(\divisor(F)\) we necessarily have \(\deg(\divisor(F))=0\), which shows that this definition is well-defined.
We claim that \(\deg\) is an isomorphism. Surjectivity of \(\deg\) is obvious because the images of the \(dH\) are \(d\). For injectivity we must show that any \(D\) with \(\deg(D)=0\) is a principal divisor. But we saw above that \(D=dH\), so 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. Thus, if a function defined on \(\mathbb{P}^n\) has a zero at some point, it must necessarily have a pole at another point, and the sum of the zero orders must equal the sum of the pole orders. The statement \(\Cl(\mathbb{P}^n)\cong \mathbb{Z}\) means that for \(d\neq 0\), the divisors \(dH\) are essentially all the non-principal divisors of \(\mathbb{P}^n\), while the remaining divisors \(\sum n_i D_i\) merely distribute this order \(d\) to the individual \(D_i\) via a rational function \(f\).
Cartier Divisors
A Weil divisor is geometrically intuitive, but it has the drawback of not working well on a singular variety. For example, on the cone \(X=Z(\x^2 + \y^2 - \z^2) \subset \mathbb{A}^3\), imagine trying to define 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 in a neighborhood of this point by the local equation \(\y=0\), so for any given rational function \(f\) we can extract the zero and pole information along \(\y\). The problem arises at the singular point. Consider, for instance, the codimension \(1\) subvariety \(L=(t,0,t)\) in a neighborhood of the origin \((0,0,0)\). To describe this we need both equations \(\y=0\) and \(\x-\z=0\), which means we cannot define \(v_L(f)\) properly. More precisely, \(\mathcal{O}_{X,(0,0,0)}\) fails to be a (one-dimensional) regular local ring and hence is not a discrete valuation ring, so we cannot define a valuation in the first place.
Thus, the core issue is that on a singular variety there can appear codimension \(1\) subvarieties that are not expressed locally by a single equation. Therefore we simply restrict our attention to those objects that are locally expressed by a single equation. As usual, we define this by means of an appropriate kind of gluing.
Definition 12 A Cartier divisor on a variety \(X\) is the following data:
\[\{(U_i, f_i)\}_{i \in I}\]where \(\{U_i\}\) is an open cover of \(X\), each \(f_i \in \mathbb{K}(X)^\ast\) is a nonzero rational function, and for all \(i, j\) the quotient \(f_i/f_j\) is regular and non-vanishing (hence invertible) on \(U_i \cap U_j\).
Two such data \(\{(U_i, f_i)\}\) and \(\{(V_j, g_j)\}\) are regarded as representing the same Cartier divisor if there exists a common refinement \(\{W_k\}\) of the two open covers on which \(f_i/g_j\) is regular and non-vanishing.
By definition a Cartier divisor is nothing but a collection of codimension 1 subvarieties that are locally principal. The data \(U_i\) explicitly specifies exactly which open set we are localizing to, and by which single function it is defined on that open set. Consequently, the line \(L\) examined above cannot be expressed by a single equation on any open set containing the origin, so it is not a Cartier divisor.
Example 13 In the example of the cone 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 can be thought of as a codimension 1 subvariety with the additional locally principal condition. Concretely, 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 therefore we can glue these together to define a Weil divisor.
We saw above that the reverse direction does not hold in general; at the same time, in the smooth case the stalk becomes a regular local ring, so there is no problem. 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 the role of a specific subset of the space \(X\), analogous to a homology class, whereas a Cartier divisor plays the role of functions defined (locally) on \(X\), analogous to a cohomology class. From this viewpoint, the proposition may be thought of 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. For this 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 8.
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 \(\CaCl(X) \cong \Cl(X)\).
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.
댓글남기기