Under a constant anti-canonical volume condition on a Zariski-dense set of Fano type fibers, the geometric generic fiber is Fano type; the dimension-2 case is unconditional.
The volume function is upper semicontinuous on families of divisors
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abstract
We study the behavior of volumes of divisors in a family. We show that the volume of a divisor on the generic fiber equals the infimum of its volumes on fibers over any dense subset of the base. As an application, we show that the volume function of a divisor is upper semicontinuous in flat families with reduced and irreducible fibers.
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On volumes and the generic invariance of Fano type varieties
Under a constant anti-canonical volume condition on a Zariski-dense set of Fano type fibers, the geometric generic fiber is Fano type; the dimension-2 case is unconditional.