On behavior of conductors, Picard schemes, and Jacobian numbers of varieties over imperfect fields

In this paper, we study invariants and related phenomena of regular but not necessarily geometrically varieties rings over imperfect fields, as exemplified by Tate's genus change theorems. particular, (i) give a geometric-normality criterion such rings, (ii) the Picard schemes curves, (iii) define new relating to δ-invariants, changes, conductors, Jacobian numbers. As an application (iii), refinements theorem [18, Theorem 1.2], show that number curve is 2p/(p−1) times change.