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

Abstract

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