  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.
