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.
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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.