Let $R$ be a complete Henselian discrete valuation ring, $\pi:X \to \mathrm{Spec} (R)$ be a smooth, proper, integral, flat $\mathrm{Spec} (R)$-scheme of dimension $2$. Assume that the genus of the special fiber (which is a smooth curve) is at least $2$. Is there any condition on $X$ such that $H^1(\mathcal{O}_X)=0$?
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$\begingroup$ What kind of conditions are you looking for? Conditions on the generic fibre? At the moment your question seems too vague to answer. $\endgroup$– Daniel LoughranCommented Jul 11, 2014 at 15:50
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$\begingroup$ @Loughran: The question is vague because I am not sure which is a good place to start looking for. $\endgroup$– user46578Commented Jul 11, 2014 at 16:44
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1 Answer
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Let $s,\eta $ be the closed and generic points of $\mathrm{Spec}(R)$. Since you assume $X$ smooth over $R$, we have $h^0(\mathscr{O}_{X_\eta })=h^0(\mathscr{O}_{X_s})=1$, hence $h^1(\mathscr{O}_{X_\eta })=h^1(\mathscr{O}_{X_s})\geq 2$. Thus $H^1(X,\mathscr{O}_X)$ is a $R$-module of rank $\geq 2$, it cannot be 0.
