This analysis demonstrates the invariance of Fano type properties in fibers, suggesting implications for volume constancy in varieties.
We demonstrate the generic invariance of the Fano type property in cases where the volumes of anti-canonical divisors of Fano type fibers are a constant over a Zariski-dense subset, or the Fano type fibers are dimension $2$. Additionally, paralleling this theorem, we establish a conjecture by Schwede and Smith under the condition that the volumes of anti-canonical divisors remain constant in the reduction mod p.
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Donghyeon Kim (2025) studied this question.
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