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

TL;DR

This paper investigates how the behavior of conductors, Picard schemes, and Jacobian numbers of varieties over imperfect fields differs from the characteristic zero case, introducing new invariants and refining classical theorems.

Contribution

It defines new invariants for local rings at codimension 1 points over imperfect fields and applies them to refine genus change theorems and relate Jacobian numbers to genus changes.

Findings

Refined Tate's genus change theorem.

Established relation between Jacobian number and genus change.

Connected Picard scheme structure with singular point invariants.

Abstract

Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\times_{\mathrm{Spec}\,K}\mathrm{Spec}\,K'$ may have singular points for a (necessarily inseparable) field extension $K'/K$. In this paper, we define new invariants of the local rings of codimension $1$ points of $X'$, and use these invariants for the calculation of $\delta$-invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and the Patakfalvi-Waldron Theorem. Moreover, when $X$ is a curve, we show that the Jacobian number of $X$ is $2p/(p-1)$ times of the genus change by using the above calculation. In this case, we also relate the structure of the Picard scheme of $X$ with invariants of singular points of $X$. To prove such a relation, we give a characterization of the geometrical normality of algebras over fields of positive characteristic.