# Fibered surfaces degenerating to Frobenius

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 $$F^e:C_k\rightarrow C_k$$?