Let $R$ be a DVR of mixed characteristic, with algebraically closed residue field of characteristic $p$ and fraction field $K$. Let $Y\longrightarrow \operatorname{Spec} R$ be a smooth projective morphism of relative dimension 2. Let $f:Y\longrightarrow \mathbb{P}_R^1$ be a morphism admitting a smooth section, such that $f_K: Y_K\longrightarrow\mathbb{P}_K^1$ and $f_k: Y_k\longrightarrow\mathbb{P}_k^1$ are elliptic surfaces of Kodaira dimension one, whose only singular fibers are of type $I_m$ ($m$ is allowed to vary across the singular fibers). Is there an example of $Y$ as above so that, furthermore, the specialization map $$\operatorname{Pic}^\tau(Y_K/\mathbb{P}^1_K)[p]\longrightarrow \operatorname{Pic}^\tau(Y_k/\mathbb{P}^1_k)[p]$$ is not injective?
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$\begingroup$ What does the $\tau$ mean here? $\endgroup$– Will SawinJan 28, 2021 at 3:42
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$\begingroup$ @WillSawin sorry, $\tau$ means torsion (somewhat redundantly) $\endgroup$– pozioJan 28, 2021 at 5:52
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1$\begingroup$ Are you sure $p$-torsion in the Picard group can exist at all? Any line bundle whose restriction to the generic fiber is degree $0$ can be written over the generic fiber as a section minus the zero section, hence is a section minus the zero section plus a bunch of divisors restricted to the fibers. Because it's torsion, it has zero intersection with each divisor on the fibers, so it's a section minus the zero section plus a bunch of copies of the generic fiber, and thus the section must intersect each fiber in the identity component. $\endgroup$– Will SawinJan 28, 2021 at 19:53
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$\begingroup$ But I don't think a $p$-torsion section in characteristic $p$ can intersect an $I_n$ fiber in the identity component (because then it would have to intersect $0$). $\endgroup$– Will SawinJan 28, 2021 at 19:58
1 Answer
As Will Sawin points out, there is no $p$-torsion in the Picard in the first place. Here is my attempt of an answer.
Let $f_{\bar{K}}:Y_{\bar{K}}\longrightarrow\mathbb{P}_{\bar{K}}^1$ be the restriction of $f$ over the geometric generic point of $\mathrm{Spec}R$. By the Lefschetz principle, we may assume that everything is defined over $\mathbb{C}$. Let $S\subset Y_{\bar{K}}$ be the smooth section of $f_{\bar{K}}$ and suppose $L\in\mathrm{Pic}(Y_{\bar{K}}/\mathbb{P}_{\bar{K}}^1)[p]$ is a nontrivial line bundle. Then $L^p=f^\ast\mathcal{O}_{\mathbb{P}^1_{\bar{K}}}(n)$ for some $n\in \mathbb{Z}$. Intersecting with $S$ we have $n\in p\mathbb{Z}$. Replacing $L$ with $L\otimes f^\ast\mathcal{O}_{\mathbb{P}^1_{\bar{K}}}(-n/p)$ we have a nontrivial $L\in\mathrm{Pic}(Y_{\bar{K}})[p]$, thus a nontrivial étale cover. On the other hand, since $\kappa(X_{\bar{K}})=1$, we have $\chi(\mathcal{O}_{X_\bar{K}})>0$, hence $\pi_1(Y_{\bar{K}})\cong\pi_1^{\mathrm{orb}}(\mathbb{P}^1_{\bar{K}})$ by Theorem 2.3 in Chapter II Section 2 of this book. As there are no multiple fibers, we have $\pi_1^{\mathrm{orb}}(\mathbb{P}^1_{\bar{K}})=0$, a contradiction.