# Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of characteristic $p>0$, the special fiber $X_k:=X_R \times_R \mathrm{Spec}(k)$ is a smooth projective curve and $\mathcal{O}_{X_k}(1)$ is the associated very ample line bundle. Assume that there exists a line bundle $L$ on $X_R$ such that its pull-back to $X_k$ is isomorphic to the very ample line bundle $\mathcal{O}_{X_k}(1)$. Is there any condition on $X_R$, under which such an $L$ is unique i.e., if there exists a line bundle $L'$ on $X_R$ which pulls back to $\mathcal{O}_{X_k}(1)$ then $L'$ is isomorphic to $L$?

• I think it should be enough that $H^1(X_k, \mathcal{O})=0$, right? – Daniel Litt Jul 5 '14 at 8:51
• @Litt: Thanks for the comment. I think thats true. But this is saying that $X_k$ is a rational curve. Is there any more general condition? – Kali Jul 5 '14 at 9:14
• Ah sorry, I didn't see that you said $X$ was an $R$-curve. In this case the answer is that this is the case if and only if $H^1(X_k, \mathcal{O})=0$. Otherwise there are many deformations of $L$ (they are parametrized by the local ring of the point on $\text{Pic}(X/R)$ corresponding to $L$), whose tangent space is $\text{Ext}^1(L, L)=H^1(X_k, \mathcal{O})$. – Daniel Litt Jul 5 '14 at 15:59