Flat projective morphisms between characteristic-zero varieties with rational singularities have locally free higher direct images of O_X and of the relative dualizing sheaf, making Pic^0(X/S) smooth.
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Higher direct images of dualizing sheaves III
Flat projective morphisms between characteristic-zero varieties with rational singularities have locally free higher direct images of O_X and of the relative dualizing sheaf, making Pic^0(X/S) smooth.