Let $S $ be the spectrum of a complete dvr with algebraically closed residue field. Let $\eta$ be its generic point and let $s$ be its closed point with $k(s)$ of positive characteristic. Let $X\to S$ be a smooth proper morphism of schemes whose geometric fibres are connected.

Edit: For all $b$ in $S$, assume that $\mathrm{H}^1(X_b,\mathcal{O}_{X_b}) = \mathrm{H}^0(X_b,\Omega^1_{X_b})=0$ and that $\mathrm{Pic}(X_b)$ is torsion-free. Moreover, assume that $\mathrm{rk}(\mathrm{Pic}(X_{\eta})) = \mathrm{rk}(\mathrm{Pic}(X_s))$. Also, assume the fibres of $X\to S$ are Fano varieties. (Some of these assumptions are redundant.)

Let $L$ be a line bundle on $X_s$ such that $L^{\otimes p}\cong \omega_{X_s/k(s)}$, where $p$ equals the characteristic of $k(s)$.

Question. Does $L$ lift to $X$?

The motivation for this question comes from a question about the index of a Fano variety. Namely, it seems reasonable to suspect that the index of a Fano variety doesn't jump upon specialization. This is clear in characteristic zero, as $\mathrm{H}^1(X,\mathcal{O}_X)=\mathrm{H}^2(X,\mathcal{O}_X)=0$ for a Fano variety $X$ over $\mathbb C$. However, the vanishing of these cohomology groups is currently not known for Fano varieties over fields of positive characteristic. (The vanishing of $\mathrm{H}^1(X,\mathcal{O}_X)$ is known for SRC Fano varieties.)

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    $\begingroup$ Are you allowing a ramified extension of $S$? I believe there are specializing families of Enriques surfaces in characteristic $2$ such that the relative $\text{Pic}^\tau$ is not smooth over $S$, although it is flat over $S$. So for such a family, you woud need to allow a ramified extension of $S$. $\endgroup$ – Jason Starr Feb 21 '17 at 16:56
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    $\begingroup$ The canonical bundle is a "red herring" (in my opinion). Let $M$ be any invertible sheaf on $X$, and let $r>1$ be any integer. Let $E$ be the locally free sheaf of rank $r$, $E=(M\otimes_{\mathcal{O}_X}\omega_{X/S}^{\vee})\oplus \mathcal{O}_X^{\oplus (r-1)}$. Let $\pi:\mathbb{P}_X(E)\to X$ and $q:\pi^*E \to \mathcal{O}(1)$ be the associated projective bundle with its natural invertible quotient. Then every $r^{\text{th}}$ root of $\omega_{\mathbb{P}_X(E)/S}$ on $\mathbb{P}_X(E)$ is of the form $\pi^*L\otimes \mathcal{O}(-1)$ for a unique $r^{\text{th}}$ root $L$ of $M$ on $X$. $\endgroup$ – Jason Starr Feb 21 '17 at 17:44
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    $\begingroup$ Also, using SGA2, you might as well assume that the relative dimension of $X\to S$ equals $2$. The problem should be birational, so presumably you can also assume that there exists a "Lefschetz fibration", $X\to \mathbb{P}^1_S \to S$. On the dense open $U\subset \mathbb{P}^1_S$ over which this morphism is smooth, you have the $r$-torsion in the relative Picard scheme. You have a torsor for this group scheme over the closed fiber, and you are trying to lift that torsor . . . $\endgroup$ – Jason Starr Feb 21 '17 at 17:55
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    $\begingroup$ For fixed invertible sheaf $L_e$ on $X\times_S \text{Spec}\mathcal{O}_{S,s}/\mathfrak{m}^{e+1}$ it looks to me lie the obstruction class $o(e,L_e)$ in $H^2(X_s,\mathcal{O}_{X_s})\otimes \mathfrak{m}_{S,s}^{e+1}/\mathfrak{m}_{S,s}^{e+2}$ to lifting to an invertible sheaf on $X\times_S \text{Spec}\mathcal{O}_{S,s}/\mathfrak{m}_{S,s}^{e+2}$ satisfies $o(e,L^{\otimes n}_e) = n o(e,L_e)$. If $n$ is prime to the characteristic, then $o(e,L_e)$ equals $0$ if and only if $no(e,L_e)$ is zero. By the hypothesis that $M$ extends, it then follows that also $L_e$ extends. $\endgroup$ – Jason Starr Feb 21 '17 at 21:22
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    $\begingroup$ The infinitesimal deformation theory has to break down when the characteristic divides $n$ because of the Enriques surface examples: the relative Picard scheme is not smooth. I vaguely remember that Torsten Ekedahl studied the case when the characteristic divides $n$, but everytime I say those words, somebody shows me that I misunderstood Ekedahl's work. $\endgroup$ – Jason Starr Feb 22 '17 at 13:51

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