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Let $f:X\to Y$ be a surjective morphism of connected smooth projective varieties over an algebraically closed field.

Assume all fibers are connected smooth and none are uniruled. Is $f$ flat?

In particular if all fibers are abelian varieties is $f$ flat? Can the general fiber be e.g. a point and the special fiber be an elliptic curve?

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  • $\begingroup$ All fibers are smooth, or the generic fiber is smooth? $\endgroup$
    – AmorFati
    Commented Aug 29, 2020 at 6:43
  • $\begingroup$ @AmorFati all fibers are smooth $\endgroup$
    – Nguyen
    Commented Aug 29, 2020 at 6:52
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    $\begingroup$ "Can the general fiber be e.g. a point and the special fiber be an elliptic curve?" Welcome new contributor. That certainly cannot happen. Abyhankhar proved that for a birational morphism to a smooth target the irreducible components of fibers are point or ruled varieties. That suggests an approach to your general question. Consider the rational transformation from $Y$ to the Chow variety of $X$ determined by the fibers of $f$. If the morphism is not flat, then the closure of the graph of this rational transformation has some uniruled fibers . . . $\endgroup$
    – Jason Starr
    Commented Aug 29, 2020 at 12:42

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