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Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and 1 respectively. Is there an example of a morphism of $R$-schemes $f:X\rightarrow C$ such that $f_{\ast}\mathcal{O}_{X}=\mathcal{O}_{C}$ and on the central fiber $f_k$ factors through an iterate of the $k$-linear Frobenius $F^e:C_k\rightarrow C_k$?

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