## 110.49 Divisors

We collect all relevant definitions here in one spot for convenience.

Definition 110.49.1. Throughout, let $S$ be any scheme and let $X$ be a Noetherian, integral scheme.

1. A Weil divisor on $X$ is a formal linear combination $\Sigma n_ i[Z_ i]$ of prime divisors $Z_ i$ with integer coefficients.

2. A prime divisor is a closed subscheme $Z \subset X$, which is integral with generic point $\xi \in Z$ such that ${\mathcal O}_{X, \xi }$ has dimension $1$. We will use the notation ${\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi }$ when $\xi \in Z \subset X$ is as above. Note that ${\mathcal O}_{X, Z} \subset K(X)$ is a subring of the function field of $X$.

3. The Weil divisor associated to a rational function $f \in K(X)^\ast$ is the sum $\Sigma v_ Z(f)[Z]$. Here $v_ Z(f)$ is defined as follows

1. If $f \in {\mathcal O}_{X, Z}^\ast$ then $v_ Z(f) = 0$.

2. If $f \in {\mathcal O}_{X, Z}$ then

$v_ Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f)).$
3. If $f = \frac{a}{b}$ with $a, b \in {\mathcal O}_{X, Z}$ then

$v_ Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)).$
4. An effective Cartier divisor on a scheme $S$ is a closed subscheme $D \subset S$ such that every point $d\in D$ has an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset S$ in $S$ so that $D \cap U = \mathop{\mathrm{Spec}}(A/(f))$ with $f \in A$ a nonzerodivisor.

5. The Weil divisor $[D]$ associated to an effective Cartier divisor $D \subset X$ of our Noetherian integral scheme $X$ is defined as the sum $\Sigma v_ Z(D)[Z]$ where $v_ Z(D)$ is defined as follows

1. If the generic point $\xi$ of $Z$ is not in $D$ then $v_ Z(D) = 0$.

2. If the generic point $\xi$ of $Z$ is in $D$ then

$v_ Z(D) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f))$

where $f \in {\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi }$ is the nonzerodivisor which defines $D$ in an affine neighbourhood of $\xi$ (as in (4) above).

6. Let $S$ be a scheme. The sheaf of total quotient rings ${\mathcal K}_ S$ is the sheaf of ${\mathcal O}_ S$-algebras which is the sheafification of the pre-sheaf ${\mathcal K}'$ defined as follows. For $U \subset S$ open we set ${\mathcal K}'(U) = S_ U^{-1}{\mathcal O}_ S(U)$ where $S_ U \subset {\mathcal O}_ S(U)$ is the multiplicative subset consisting of sections $f \in {\mathcal O}_ S(U)$ such that the germ of $f$ in ${\mathcal O}_{S, u}$ is a nonzerodivisor for every $u\in U$. In particular the elements of $S_ U$ are all nonzerodivisors. Thus ${\mathcal O}_ S$ is a subsheaf of ${\mathcal K}_ S$, and we get a short exact sequence

$0 \to {\mathcal O}_ S^\ast \to {\mathcal K}_ S^\ast \to {\mathcal K}_ S^\ast /{\mathcal O}_ S^\ast \to 0.$
7. A Cartier divisor on a scheme $S$ is a global section of the quotient sheaf ${\mathcal K}_ S^\ast /{\mathcal O}_ S^\ast$.

8. The Weil divisor associated to a Cartier divisor $\tau \in \Gamma (X, {\mathcal K}_ X^\ast /{\mathcal O}_ X^\ast )$ over our Noetherian integral scheme $X$ is the sum $\Sigma v_ Z(\tau )[Z]$ where $v_ Z(\tau )$ is defined as by the following recipe

1. If the germ of $\tau$ at the generic point $\xi$ of $Z$ is zero – in other words the image of $\tau$ in the stalk $({\mathcal K}^\ast /{\mathcal O}^\ast )_\xi$ is “zero” – then $v_ Z(\tau ) = 0$.

2. Find an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ so that $\tau |_ U$ is the image of a section $f \in {\mathcal K}(U)$ and moreover $f = a/b$ with $a, b \in A$. Then we set

$v_ Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)).$

Remarks 110.49.2. Here are some trivial remarks.

1. On a Noetherian integral scheme $X$ the sheaf ${\mathcal K}_ X$ is constant with value the function field $K(X)$.

2. To make sense out of the definitions above one needs to show that

$\text{length}_{\mathcal O}({\mathcal O}/(ab)) = \text{length}_{\mathcal O}({\mathcal O}/(a)) + \text{length}_{\mathcal O}({\mathcal O}/(b))$

for any pair $(a, b)$ of nonzero elements of a Noetherian 1-dimensional local domain ${\mathcal O}$. This will be done in the lectures.

Exercise 110.49.3. (On any scheme.) Describe how to assign a Cartier divisor to an effective Cartier divisor.

Exercise 110.49.4. (On an integral scheme.) Describe how to assign a Cartier divisor $D$ to a rational function $f$ such that the Weil divisor associated to $D$ and to $f$ agree. (This is silly.)

Exercise 110.49.5. Give an example of a Weil divisor on a variety which is not the Weil divisor associated to any Cartier divisor.

Exercise 110.49.6. Give an example of a Weil divisor $D$ on a variety which is not the Weil divisor associated to any Cartier divisor but such that $nD$ is the Weil divisor associated to a Cartier divisor for some $n > 1$.

Exercise 110.49.7. Give an example of a Weil divisor $D$ on a variety which is not the Weil divisor associated to any Cartier divisor and such that $nD$ is NOT the Weil divisor associated to a Cartier divisor for any $n > 1$. (Hint: Consider a cone, for example $X : xy - zw = 0$ in $\mathbf{A}^4_ k$. Try to show that $D = [x = 0, z = 0]$ works.)

Exercise 110.49.8. On a separated scheme $X$ of finite type over a field: Give an example of a Cartier divisor which is not the difference of two effective Cartier divisors. Hint: Find some $X$ which does not have any nonempty effective Cartier divisors for example the scheme constructed in [III Exercise 5.9, H]. There is even an example with $X$ a variety – namely the variety of Exercise 110.49.9.

Exercise 110.49.9. Example of a nonprojective proper variety. Let $k$ be a field. Let $L \subset \mathbf{P}^3_ k$ be a line and let $C \subset \mathbf{P}^3_ k$ be a nonsingular conic. Assume that $C \cap L = \emptyset$. Choose an isomorphism $\varphi : L \to C$. Let $X$ be the $k$-variety obtained by glueing $C$ to $L$ via $\varphi$. In other words there is a surjective proper birational morphism

$\pi : \mathbf{P}^3_ k \longrightarrow X$

and an open $U \subset X$ such that $\pi : \pi ^{-1}(U) \to U$ is an isomorphism, $\pi ^{-1}(U) = \mathbf{P}^3_ k \setminus (L \cup C)$ and such that $\pi |_ L = \pi |_ C \circ \varphi$. (These conditions do not yet uniquely define $X$. In order to do this you need to specify the structure sheaf of $X$ along points of $Z = X \setminus U$.) Show $X$ exists, is a proper variety, but is not projective. (Hint: For existence use the result of Exercise 110.37.9. For non-projectivity use that $\mathop{\mathrm{Pic}}\nolimits (\mathbf{P}^3_ k) = \mathbf{Z}$ to show that $X$ cannot have an ample invertible sheaf.)

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