The Stacks project

Lemma 42.41.4. Let $k$ be a field. Let $X$ be a proper scheme 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. Then

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = \deg ( c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [Z]_ d) \]

where the left hand side is defined in Varieties, Definition 33.45.3. In particular,

\[ \deg _\mathcal {L}(Z) = \deg (c_1(\mathcal{L})^ d \cap [Z]_ d) \]

if $\mathcal{L}$ is an ample invertible $\mathcal{O}_ X$-module.

Proof. We will prove this by induction on $d$. If $d = 0$, then the result is true by Varieties, Lemma 33.33.3. Assume $d > 0$.

Let $Z_ i \subset Z$, $i = 1, \ldots , t$ be the irreducible components of dimension $d$ with reduced induced scheme structure and let $m_ i$ be the multiplicity of $Z_ i$ in $Z$. Then $[Z]_ d = \sum m_ i[Z_ i]$ and $c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [Z]_ d$ is the sum of the cycles $m_ i c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [Z_ i]$. Since we have a similar decomposition for $(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z)$ by Varieties, Lemma 33.45.2 it suffices to prove the lemma in case $Z = X$ is a proper variety of dimension $d$ over $k$.

By Chow's lemma there exists a birational proper morphism $f : Y \to X$ with $Y$ H-projective over $k$. See Cohomology of Schemes, Lemma 30.18.1 and Remark 30.18.2. Then

\[ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_ d \cdot Y) = (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot X) \]

by Varieties, Lemma 33.45.7 and we have

\[ f_*(c_1(f^*\mathcal{L}_1) \cap \ldots \cap c_1(f^*\mathcal{L}_ d) \cap [Y]) = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [X] \]

by Lemma 42.26.4. Thus we may replace $X$ by $Y$ and assume that $X$ is projective over $k$.

If $X$ is a proper $d$-dimensional projective variety, then we can write $\mathcal{L}_1 = \mathcal{O}_ X(D) \otimes \mathcal{O}_ X(D')^{\otimes -1}$ for some effective Cartier divisors $D, D' \subset X$ by Divisors, Lemma 31.15.12. By additivity for both sides of the equation (Varieties, Lemma 33.45.5 and Lemma 42.25.2) we reduce to the case $\mathcal{L}_1 = \mathcal{O}_ X(D)$ for some effective Cartier divisor $D$. By Varieties, Lemma 33.45.8 we have

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot X) = (\mathcal{L}_2 \cdots \mathcal{L}_ d \cdot D) \]

and by Lemma 42.25.4 we have

\[ c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [X] = c_1(\mathcal{L}_2) \cap \ldots \cap c_1(\mathcal{L}_ d) \cap [D]_{d - 1} \]

Thus we obtain the result from our induction hypothesis. $\square$


Comments (4)

Comment #5561 by Dario Weißmann on

typo in the statement: deg should be deg

Comment #7937 by 11k on

Is ampleness of redundant?


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BFI. Beware of the difference between the letter 'O' and the digit '0'.