A write-up of Moret-Bailly's criteria: over a henselian DVR, a locally finite type morphism with integral special fibre is flat and surjective exactly when algebraic special-fibre points lift to points over the integral closure.
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Surjectivity and flatness over DVR's (after Moret-Bailly)
A write-up of Moret-Bailly's criteria: over a henselian DVR, a locally finite type morphism with integral special fibre is flat and surjective exactly when algebraic special-fibre points lift to points over the integral closure.