Let $X$ be a smooth projective variety over a field $k$ of characteristic zero, and let $D$ be a simple normal crossing divisor inside $X$ having irreducible components $D_i$. Further let $x \in X$ be a closed point belonging to (at least) one of these components, say $D_0$.

How should I think of the elements in

$\Gamma(Spec(\mathcal{O}_{X, x})-D_0, \mathbb{G}_m)$?

What is the order at $x$ of such an element?

Thanks for your help!


In $\mathrm{Spec}(\mathcal{O}_{X,x})$, $D_0$ is given by one equation $f=0$. So $\Gamma(Spec(\mathcal{O}_{X, x})-D_0, \mathbb{G}_m)$ is just $\mathcal{O}_{X, x}^*[\dfrac{1}{f} ]$: that is, any element is of the form $uf^n$, where $u$ is invertible near $x$ and $n\in \Bbb{Z}$. Its order at $x$ is $n$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.