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.
N\'eron models, minimal models, and birational group actions
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abstract
Let $A_K$ be an Abelian variety over the quotient field of a Dedekind domain $R$. We show that the identity component of the N\'eron model of $A_K$ acts regularly on any minimal model of $A_K$ over $R$, and discuss when the N\'eron model is an open subset of a minimal model. The main technical result is the rigidity of small modifications, leading to a regularity criterion for birational group actions.
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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.