On volumes and the generic invariance of Fano type varieties

TL;DR

This paper proves the invariance of Fano type properties under certain conditions related to volumes of anti-canonical divisors, advancing understanding in algebraic geometry and singularity theory.

Contribution

It establishes the generic invariance of Fano type properties when volumes of anti-canonical divisors are constant, and confirms a conjecture by Schwede and Smith in specific cases.

Findings

Invariance of Fano type under volume constancy over Zariski-dense sets

Validation of Schwede and Smith's conjecture in characteristic p reductions

Extension of invariance results to Fano type fibers of dimension 2

Abstract

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 $p$.