On volumes and the generic invariance of Fano type varieties

Key Points

• The findings show that the volumes of anti-canonical divisors remain constant in certain Fano type varieties, impacting their classification.
• The research confirms a conjecture by Schwede and Smith, linking volume constancy and the dimensional characteristics of Fano type fibers.
• The study explores cases of Fano type fiber varieties where the Zariski-dense subset condition holds, offering new insights into their invariance properties.
• This work emphasizes the relevance of anti-canonical divisors under reduction mod p, expanding the understanding of Fano varieties.