Suppose I have a morphism of Noetherian schemes over a field $k$ (if one needs this then assume $k$ is algebraically closed) $f:C'\rightarrow S$ which is finite with geometrically connected and reduced fibers. Is $f$ an isomorphism?



Does finite+reduced fibers+connected fibers imply isomorphism?
No: $\text{Spec} k[\epsilon]/\langle \epsilon \rangle$ mapping to $\text{Spec} k[\epsilon]/\langle \epsilon^2 \rangle$.


