
32.44 Numerical intersections

In this section we play around with the Euler characteristic of coherent sheaves on proper schemes to obtain numerical intersection numbers for invertible modules. Our main tool will be the following lemma.

Lemma 32.44.1. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ r$ be invertible $\mathcal{O}_ X$-modules. The map

$(n_1, \ldots , n_ r) \longmapsto \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r})$

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree at most the dimension of the support of $\mathcal{F}$.

Proof. We prove this by induction on $\dim (\text{Supp}(\mathcal{F}))$. If this number is zero, then the function is constant with value $\dim _ k \Gamma (X, \mathcal{F})$ by Lemma 32.32.3. Assume $\dim (\text{Supp}(\mathcal{F})) > 0$.

If $\mathcal{F}$ has embedded associated points, then we can consider the short exact sequence $0 \to \mathcal{K} \to \mathcal{F} \to \mathcal{F}' \to 0$ constructed in Divisors, Lemma 30.4.6. Since the dimension of the support of $\mathcal{K}$ is strictly less, the result holds for $\mathcal{K}$ by induction hypothesis and with strictly smaller total degree. By additivity of the Euler characteristic (Lemma 32.32.2) it suffices to prove the result for $\mathcal{F}'$. Thus we may assume $\mathcal{F}$ does not have embedded associated points.

If $i : Z \to X$ is a closed immersion and $\mathcal{F} = i_*\mathcal{G}$, then we see that the result for $X$, $\mathcal{F}$, $\mathcal{L}_1, \ldots , \mathcal{L}_ r$ is equivalent to the result for $Z$, $\mathcal{G}$, $i^*\mathcal{L}_1, \ldots , i^*\mathcal{L}_ r$ (since the cohomologies agree, see Cohomology of Schemes, Lemma 29.2.4). Applying Divisors, Lemma 30.4.7 we may assume that $X$ has no embedded components and $X = \text{Supp}(\mathcal{F})$.

Pick a regular meromorphic section $s$ of $\mathcal{L}_1$, see Divisors, Lemma 30.23.13. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal of denominators of $s$ and consider the maps

$\mathcal{I}\mathcal{F} \to \mathcal{F},\quad \mathcal{I}\mathcal{F} \to \mathcal{F} \otimes \mathcal{L}_1$

of Divisors, Lemma 30.23.16. These are injective and have cokernels $\mathcal{Q}$, $\mathcal{Q}'$ supported on nowhere dense closed subschemes of $X = \text{Supp}(\mathcal{F})$. Tensoring with the invertible module $\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_1^{\otimes n_1}$ is exact, hence using additivity again we see that

\begin{align*} & \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1 + 1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) \\ & = \chi (\mathcal{Q} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \chi (\mathcal{Q}' \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) \end{align*}

Thus we see that the function $P(n_1, \ldots , n_ r)$ of the lemma has the property that

$P(n_1 + 1, n_2, \ldots , n_ r) - P(n_1, \ldots , n_ r)$

is a numerical polynomial of total degree $<$ the dimension of the support of $\mathcal{F}$. Of course by symmetry the same thing is true for

$P(n_1, \ldots , n_{i - 1}, n_ i + 1, n_{i + 1}, \ldots , n_ r) - P(n_1, \ldots , n_ r)$

for any $i \in \{ 1, \ldots , r\}$. A simple arithmetic argument shows that $P$ is a numerical polynomial of total degree at most $\dim (\text{Supp}(\mathcal{F}))$. $\square$

The following lemma roughly shows that the leading coefficient only depends on the length of the coherent module in the generic points of its support.

Lemma 32.44.2. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ r$ be invertible $\mathcal{O}_ X$-modules. Let $d = \dim (\text{Supp}(\mathcal{F}))$. Let $Z_ i \subset X$ be the irreducible components of $\text{Supp}(\mathcal{F})$ of dimension $d$. Let $\xi _ i \in Z_ i$ be the generic point and set $m_ i = \text{length}_{\mathcal{O}_{X, \xi _ i}}(\mathcal{F}_{\xi _ i})$. Then

$\chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \sum \nolimits _ i m_ i\ \chi (Z_ i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_{Z_ i})$

is a numerical polynomial in $n_1, \ldots , n_ r$ of total degree $< d$.

Proof. Consider pairs $(\xi , Z)$ where $Z \subset X$ is an integral closed subscheme of dimension $d$ and $\xi$ is its generic point. Then the finite $\mathcal{O}_{X, \xi }$-module $\mathcal{F}_\xi$ has support contained in $\{ \xi \}$ hence the length $m_ Z = \text{length}_{\mathcal{O}_{X, \xi }}(\mathcal{F}_\xi )$ is finite (Algebra, Lemma 10.61.3) and zero unless $Z = Z_ i$ for some $i$. Thus the expression of the lemma can be written as

$E(\mathcal{F}) = \chi (X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) - \sum \nolimits m_ Z\ \chi (Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_ Z)$

where the sum is over integral closed subschemes $Z \subset X$ of dimension $d$. The assignment $\mathcal{F} \mapsto E(\mathcal{F})$ is additive in short exact sequences $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ of coherent $\mathcal{O}_ X$-modules whose support has dimension $\leq d$. This follows from additivity of Euler characteristics (Lemma 32.32.2) and additivity of lengths (Algebra, Lemma 10.51.3). Let us apply Cohomology of Schemes, Lemma 29.12.3 to find a filtration

$0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ m = \mathcal{F}$

by coherent subsheaves such that for each $j = 1, \ldots , m$ there exists an integral closed subscheme $V_ j \subset X$ and a sheaf of ideals $\mathcal{I}_ j \subset \mathcal{O}_{V_ j}$ such that

$\mathcal{F}_ j/\mathcal{F}_{j - 1} \cong (V_ j \to X)_* \mathcal{I}_ j$

By the additivity we remarked upon above it suffices to prove the result for each of the subquotients $\mathcal{F}_ j/\mathcal{F}_{j - 1}$. Thus it suffices to prove the result when $\mathcal{F} = (V \to X)_*\mathcal{I}$ where $V \subset X$ is an integral closed subscheme of dimension $\leq d$. If $\dim (V) < d$ and more generally for $\mathcal{F}$ whose support has dimension $< d$, then the first term in $E(\mathcal{F})$ has total degree $< d$ by Lemma 32.44.1 and the second term is zero. If $\dim (V) = d$, then we can use the short exact sequence

$0 \to (V \to X)_*\mathcal{I} \to (V \to X)_*\mathcal{O}_ V \to (V \to X)_*(\mathcal{O}_ V/\mathcal{I}) \to 0$

The result holds for the middle sheaf because the only $Z$ occurring in the sum is $Z = V$ with $m_ Z = 1$ and because

$H^ i(X, ((V \to X)_*\mathcal{O}_ V) \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}) = H^ i(V, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ r^{\otimes n_ r}|_ V)$

by the projection formula (Cohomology, Section 20.45) and Cohomology of Schemes, Lemma 29.2.4; so in this case we actually have $E(\mathcal{F}) = 0$. The result holds for the sheaf on the right because its support has dimension $< d$. Thus the result holds for the sheaf on the left and the lemma is proved. $\square$

Definition 32.44.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $i : Z \to X$ be a closed subscheme of dimension $d$. Let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. We define the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ as the coefficient of $n_1 \ldots n_ d$ in the numerical polynomial

$\chi (X, i_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) = \chi (Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ Z)$

In the special case that $\mathcal{L}_1 = \ldots = \mathcal{L}_ d = \mathcal{L}$ we write $(\mathcal{L}^ d \cdot Z)$.

The displayed equality in the definition follows from the projection formula (Cohomology, Section 20.45) and Cohomology of Schemes, Lemma 29.2.4. We prove a few lemmas for these intersection numbers.

Lemma 32.44.4. In the situation of Definition 32.44.3 the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ is an integer.

Proof. Any numerical polynomial of degree $e$ in $n_1, \ldots , n_ d$ can be written uniquely as a $\mathbf{Z}$-linear combination of the functions ${n_1 \choose k_1}{n_2 \choose k_2} \ldots {n_ d \choose k_ d}$ with $k_1 + \ldots + k_ d \leq e$. Apply this with $e = d$. Left as an exercise. $\square$

Lemma 32.44.5. In the situation of Definition 32.44.3 the intersection number $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ is additive: if $\mathcal{L}_ i = \mathcal{L}_ i' \otimes \mathcal{L}_ i''$, then we have

$(\mathcal{L}_1 \cdots \mathcal{L}_ i \cdots \mathcal{L}_ d \cdot Z) = (\mathcal{L}_1 \cdots \mathcal{L}_ i' \cdots \mathcal{L}_ d \cdot Z) + (\mathcal{L}_1 \cdots \mathcal{L}_ i'' \cdots \mathcal{L}_ d \cdot Z)$

Proof. This is true because by Lemma 32.44.1 the function

$(n_1, \ldots , n_{i - 1}, n_ i', n_ i'', n_{i + 1}, \ldots , n_ d) \mapsto \chi (Z, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes (\mathcal{L}_ i')^{\otimes n_ i'} \otimes (\mathcal{L}_ i'')^{\otimes n_ i''} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ Z)$

is a numerical polynomial of total degree at most $d$ in $d + 1$ variables. $\square$

Lemma 32.44.6. In the situation of Definition 32.44.3 let $Z_ i \subset Z$ be the irreducible components of dimension $d$. Let $m_ i = \text{length}_{\mathcal{O}_{X, \xi _ i}}(\mathcal{O}_{Z, \xi _ i})$ where $\xi _ i \in Z_ i$ is the generic point. Then

$(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = \sum m_ i(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z_ i)$

Proof. Immediate from Lemma 32.44.2 and the definitions. $\square$

Lemma 32.44.7. Let $k$ be a field. Let $f : Y \to X$ be a morphism of proper schemes over $k$. Let $Z \subset Y$ be an integral closed subscheme of dimension $d$ and let $\mathcal{L}_1, \ldots , \mathcal{L}_ d$ be invertible $\mathcal{O}_ X$-modules. Then

$(f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_ d \cdot Z) = \deg (f|_ Z : Z \to f(Z)) (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot f(Z))$

where $\deg (Z \to f(Z))$ is as in Morphisms, Definition 28.48.8 or $0$ if $\dim (f(Z)) < d$.

Proof. The left hand side is computed using the coefficient of $n_1 \ldots n_ d$ in the function

$\chi (Y, \mathcal{O}_ Z \otimes f^*\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes f^*\mathcal{L}_ d^{\otimes n_ d}) = \sum (-1)^ i \chi (X, R^ if_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d})$

The equality follows from Lemma 32.32.5 and the projection formula (Cohomology, Lemma 20.45.2). If $f(Z)$ has dimension $< d$, then the right hand side is a polynomial of total degree $32.44.1 and the result is true. Assume$\dim (f(Z)) = d$. Let$\xi \in f(Z)$be the generic point. By dimension theory (see Lemmas 32.20.3 and 32.20.4) the generic point of$Z$is the unique point of$Z$mapping to$\xi $. Then$f : Z \to f(Z)$is finite over a nonempty open of$f(Z)$, see Morphisms, Lemma 28.48.1. Thus$\deg (f : Z \to f(Z))$is defined and in fact it is equal to the length of the stalk of$f_*\mathcal{O}_ Z$at$\xi $over$\mathcal{O}_{X, \xi }$. Moreover, the stalk of$R^ if_*\mathcal{O}_ X$at$\xi $is zero for$i > 0$because we just saw that$f|_ Z$is finite in a neighbourhood of$\xi $(so that Cohomology of Schemes, Lemma 29.9.9 gives the vanishing). Thus the terms$\chi (X, R^ if_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d})$with$i > 0$have total degree$< d$and $\chi (X, f_*\mathcal{O}_ Z \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}) = \deg (f : Z \to f(Z)) \chi (f(Z), \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_{f(Z)})$ modulo a polynomial of total degree$< d$by Lemma 32.44.2. The desired result follows.$\square$Lemma 32.44.8. Let$k$be a field. Let$X$be proper over$k$. Let$Z \subset X$be a closed subscheme of dimension$d$. Let$\mathcal{L}_1, \ldots , \mathcal{L}_ d$be invertible$\mathcal{O}_ X$-modules. Assume there exists an effective Cartier divisor$D \subset Z$such that$\mathcal{L}_1|_ Z \cong \mathcal{O}_ Z(D)$. Then $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = (\mathcal{L}_2 \cdots \mathcal{L}_ d \cdot D)$ Proof. We may replace$X$by$Z$and$\mathcal{L}_ i$by$\mathcal{L}_ i|_ Z$. Thus we may assume$X = Z$and$\mathcal{L}_1 = \mathcal{O}_ X(D)$. Then$\mathcal{L}_1^{-1}$is the ideal sheaf of$D$and we can consider the short exact sequence $0 \to \mathcal{L}_1^{\otimes -1} \to \mathcal{O}_ X \to \mathcal{O}_ D \to 0$ Set$P(n_1, \ldots , n_ d) = \chi (X, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d})$and$Q(n_1, \ldots , n_ d) = \chi (D, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_ d^{\otimes n_ d}|_ D)$. We conclude from additivity that $P(n_1, \ldots , n_ d) - P(n_1 - 1, n_2, \ldots , n_ d) = Q(n_1, \ldots , n_ d)$ Because the total degree of$P$is at most$d$, we see that the coefficient of$n_1 \ldots n_ d$in$P$is equal to the coefficient of$n_2 \ldots n_ d$in$Q$.$\square$Lemma 32.44.9. Let$k$be a field. Let$X$be proper over$k$. Let$Z \subset X$be a closed subscheme of dimension$d$. If$\mathcal{L}_1, \ldots , \mathcal{L}_ d$are ample, then$(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$is positive. Proof. We will prove this by induction on$d$. The case$d = 0$follows from Lemma 32.32.3. Assume$d > 0$. By Lemma 32.44.6 we may assume that$Z$is an integral closed subscheme. In fact, we may replace$X$by$Z$and$\mathcal{L}_ i$by$\mathcal{L}_ i|_ Z$to reduce to the case$Z = X$is a proper variety of dimension$d$. By Lemma 32.44.5 we may replace$\mathcal{L}_1$by a positive tensor power. Thus we may assume there exists a nonzero section$s \in \Gamma (X, \mathcal{L}_1)$such that$X_ s$is affine (here we use the definition of ample invertible sheaf, see Properties, Definition 27.26.1). Observe that$X$is not affine because proper and affine implies finite (Morphisms, Lemma 28.42.11) which contradicts$d > 0$. It follows that$s$has a nonempty vanishing scheme$Z(s) \subset X$. Since$X$is a variety,$s$is a regular section of$\mathcal{L}_1$, so$Z(s)$is an effective Cartier divisor, thus$Z(s)$has codimension$1$in$X$, and hence$Z(s)$has dimension$d - 1$(here we use material from Divisors, Sections 30.13 and 30.15 and from dimension theory as in Lemma 32.20.3). By Lemma 32.44.8 we have $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot X) = (\mathcal{L}_2 \cdots \mathcal{L}_ d \cdot Z(s))$ By induction the right hand side is positive and the proof is complete.$\square$Definition 32.44.10. Let$k$be a field. Let$X$be a proper scheme over$k$. Let$\mathcal{L}$be an ample invertible$\mathcal{O}_ X$-module. For any closed subscheme the degree of$Z$with respect to$\mathcal{L}$, denoted$\deg _\mathcal {L}(Z)$, is the intersection number$(\mathcal{L}^ d \cdot Z)$where$d = \dim (Z)$. By Lemma 32.44.9 the degree of a subscheme is always a positive integer. We note that$\deg _\mathcal {L}(Z) = d$if and only if $\chi (Z, \mathcal{L}^{\otimes n}|_ Z) = \frac{d}{\dim (Z)!} n^{\dim (Z)} + l.o.t$ as can be seen using that $(n_1 + \ldots + n_{\dim (Z)})^{\dim (Z)} = \dim (Z)!\ n_1 \ldots n_{\dim (Z)} + \text{other terms}$ Lemma 32.44.11. Let$k$be a field. Let$f : Y \to X$be a finite dominant morphism of proper varieties over$k$. Let$\mathcal{L}$be an ample invertible$\mathcal{O}_ X$-module. Then $\deg _{f^*\mathcal{L}}(Y) = \deg (f) \deg _\mathcal {L}(X)$ where$\deg (f)$is as in Morphisms, Definition 28.48.8. Proof. The statement makes sense because$f^*\mathcal{L}$is ample by Morphisms, Lemma 28.35.7. Having said this the result is a special case of Lemma 32.44.7.$\square$Finally we relate the intersection number with a curve to the notion of degrees of invertible modules on curves introduced in Section 32.43. Lemma 32.44.12. Let$k$be a field. Let$X$be a proper scheme over$k$. Let$Z \subset X$be a closed subscheme of dimension$\leq 1$. Let$\mathcal{L}$be an invertible$\mathcal{O}_ X$-module. Then $(\mathcal{L} \cdot Z) = \deg (\mathcal{L}|_ Z)$ where$\deg (\mathcal{L}|_ Z)$is as in Definition 32.43.1. If$\mathcal{L}$is ample, then$\deg _\mathcal {L}(Z) = \deg (\mathcal{L}|_ Z)$. Proof. This follows from the fact that the function$n \mapsto \chi (Z, \mathcal{L}|_ Z^{\otimes n})$has degree$1$and hence the leading coefficient is the difference of consecutive values.$\square$Proposition 32.44.13 (Asymptotic Riemann-Roch). Let$k$be a field. Let$X$be a proper scheme over$k$of dimension$d$. Let$\mathcal{L}$be an ample invertible$\mathcal{O}_ X$-module. Then $\dim _ k \Gamma (X, \mathcal{L}^{\otimes n}) \sim c n^ d + l.o.t.$ where$c = \deg _\mathcal {L}(X)/d!$is a positive constant. Proof. This follows from the definitions, Lemma 32.44.9, and the vanishing of higher cohomology in Cohomology of Schemes, Lemma 29.17.1.$\square$Comments (0) Post a comment Your email address will not be published. Required fields are marked. 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