Variation of cones of divisors in a family of varieties -- Fano type case

We investigate the relationship between the Fano type property on fibers over a Zariski dense subset and the global Fano type property.

We establish the invariance of Néron-Severi spaces, nef cones, effective cones, movable cones, and Mori chamber decompositions for a family of Fano type varieties after a generically finite base change.

Additionally, we show the uniform behavior of the minimal model program for this family.

These results are applied to the boundedness problem of Fano type varieties.