this is exercise 9.1.12 b) in Qing Liu's book "Algebraic geometry and Arithmetic curves".

Let $\pi:X \rightarrow S$ be an arithmetic surface with smooth and geometrically connected generic fiber $X_{\eta}$. Let $\mathcal{L} \in Pic(X)$ be such that $\mathcal{L}|_{X_\eta} \cong \mathcal{O}|_{X_\eta}$ and $\text{deg } \mathcal{L}|_{X_s} =0$ for every $ s \in S$.

One can then show that there will exist a vertical Cartier divisor $V$ (meaning its associated Weil divisor only has support in the vertical fibers) such that $\mathcal{L} \cong \mathcal{O}_X(V)$.This is part a) of the exercise, and I have no problems with this.

Part b is the following, for any closed point $s \in S,$ let $d_s$ denote the gcd of the multiplicites of the irreducible components of $X_s$. Let $d$ be the lcm of the $d_s$. Show that $dV$ is a sum of closed fibers.

I can show that $dV$ is a sum of closed fibers if $(V|_{X_s})^2=0$ (self-intersection) for all closed points $s \in S$, but I don't see why this has to be true. Does it follow from the $\text{Deg} \mathcal{L}|_{X_s} =0$ hypothesis? If so, why?

Any help would be most appreciated.


The fact that $Deg \mathcal{L}|_{X_s}=0$ indeed implies that $(V|_{X_s})^2=0$ it turns out, essentially from the definition directly.


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.