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edited Aug 29, 2020 at 6:13

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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edited Aug 28, 2020 at 15:55

Connected Non-uniruled connected smooth fibers not containing rational curvesimplies flat

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 contain rational curvesare uniruled. Is $f$ flat?

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asked Aug 28, 2020 at 11:18

Connected smooth fibers not containing rational curves

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 contain rational curves. Is $f$ flat?