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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.

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1 Answer 1

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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.

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