Let $R$ be a Henselian discrete valuation rings with algebraically closed residue field and $X$ be a regular, flat, proper $R$-scheme. Assume that the generic fiber to the natural morphism from $X$ to $\mathrm{Spec}(R)$ is smooth and the singular fiber is simple normal crossings divisor. Is it true that $H^i(\mathcal{O}_X)$ is a torsion-free $R$-module for $i>0$ i.e., for $K=\mathrm{Frac}(R)$, is the natural morphism from $H^i(\mathcal{O}_X)$ to $H^i(\mathcal{O}_X) \otimes_R K$ injective for all $i>0$? If not true in general, is there any known condition on $X$, under which this could hold true?
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$\begingroup$ If the residue field has characteristic $0$, there are positive results (Koll'ar and Shokurov, etc.). If the residue field has characteristic $p$, this can fail, e.g., for what are called "singular" Enriques surfaces (these are smooth, proper schemes). $\endgroup$– Jason StarrCommented May 14, 2017 at 9:02
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$\begingroup$ @JasonStarr Thank you. I am mainly interested in the case the generic fiber is rationally connected but the residue field is of positive characteristic. $\endgroup$– RonCommented May 14, 2017 at 9:18
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$\begingroup$ I think that can happen already for the supersingular Enriques surfaces: I believe that these are dominated by rational surfaces (thus they are even unirational, not just rationally connected). $\endgroup$– Jason StarrCommented May 14, 2017 at 10:06
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$\begingroup$ @JasonStarr Is there any known condition, we can impose on $X$ to avoid this situation? $\endgroup$– RonCommented May 14, 2017 at 10:13
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1$\begingroup$ @JasonStarr I think I got it, thanks. $\endgroup$– RonCommented May 17, 2017 at 19:54
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