intersection number

Question

I vaguely recall the following fact that I'd like to use in my research. It should be easy to see that this holds (if it does) but I can't seem to prove it.

Let $p:X\longrightarrow S$ be a (regular) arithmetic surface over a Dedekind scheme $S$.

Let $P:S\longrightarrow X$ be a section and let $\omega$ be a non-zero rational section of $\omega_{X/S}$. Let $K_{X} = \mathrm{div}(\omega) $ be the canonical divisor defined by $\omega$. (A better notation for $K_X$ would be $K_{X/S}$, maybe.)

By definition, the intersection number $(K_X,P)$ is defined as $$\sum_{s } i_s (K_{X}, P) \log( \mathrm{card}( k (s)) ), $$ where the sum runs over the closed points of $S$ and $$i_s(K_{X},P) = \sum_{x} i_x(K_{X}, P) [k(x):k(s)],$$ where the sum runs over the closed points of $X_s$ and $i_x$ denotes the intersection number at $x$.

Now, I wonder if the following equality is trivial to see.

Write $\omega = df$ for some rational function $f\in K(X)$. (We assume this to be possible.) Do we have that

$$(K_{X}, P) = \sum_{s} \mathrm{ord}_s(P^\ast\omega) \log(\mathrm{card}(k(s)))?$$

To see this, it suffices to prove the following equality: $$\mathrm{ord}_s(P^\ast \omega) =i_s (K_{X}, P) .$$ Does this hold?

Answer 1

This is correct if $P(S)$ is not contained in the support of $\mathrm{div}(\omega)$. It comes essentially from the definition of $i_x(K_X, P)$. You don't need $\omega$ to be an exact differential from. However the intersection number depends on the choice of $\omega$ (as well as the Weil divisor $K_X$). You can check this by yourself by multiplying $\omega$ by a non-zero constant in $K(S)$ and see the effect on the total intersection number. If $P(S)$ is contained in the support of $K_X$, then you can't define $i_x(K_X, P)$.

EDIT. Let me add some more details. Denote by $K(X)$ the field of rational functions on $X$, viewed as a constant sheaf on $X$. Then $\omega\in \omega_{X/S}\otimes K(X)$. Hence $\omega\cdot\omega_{X/S}^{\vee}$ is a subsheaf of $K(X)$, hence equal (not only isomorphic) to $O_X(-D)$ for some Cartier divisor $D$ on $X$. We have $$ \omega_{X/S}=\omega\cdot O_X(D). $$ A straightforward local computation shows that $\mathrm{div}(\omega)=D$ as Cartier divisors. Let us identify $P$ with $P(S)$. Let $I\subset O_X$ be the ideal sheaf defining $P$ in $X$. As $P$ is not contained in the support of $D$, $D|_P$ is a well defined Cartier divisor on $P$. Namely, if a local equation of $D$ at some point $x\in P$ is given by $f_x\in K(X)$, then we can write $f_x=a/b$ with $a, b\notin I_x$ (here we use the fact that $O_{X,x}$ is a UFD). Then a local equation of $D$ restricted to $P$ is $\bar{a}/\bar{b}$ where $\bar{c}$ means the image of $c$ in $O_{X,x}/I$.

The above equality restricted to $P$ reads $$P^{*}(\omega_{X/S})=P^{*}(\omega) \cdot O_P(D|_P).$$ So $P^{*}(\omega)$ is a rational section of

$P^{*}\omega_{X/S}$ and its divisor on $S$ is $D|_P$.

To get an intersection number independent of the choice of a rational section $\omega$, you have to use Arakelov intersection theory.