Ramification of surfaces: Artin-Schreier extensions

Let $A$ be a regular 2-dimensional local ring of characteristic $p>0$, and let $L\mathrm{/}K$ be a cyclic extension of degree $p$ of its field of fractions such that the corresponding branch divisor is normal crossing. For each $\gp\in\Spec A$ofheight1suchthat$A/\gp$isregular,considertheramificationjump$h_\gp$oftheextensionoftheresiduefieldat$\gp$. In this paper the semi-continuity of $h_{\text{gp}}$ with respect to Zariski topology of suitable jet spaces is proved. The asymptotic of $h_{\text{gp}}$ with respect to intersection multiplicity of the prime divisor defined by $\text{gp}$ and the branch divisor is also addressed.