Jet schemes of singular surfaces of types $D_4^0$ and $D_4^1$ in characteristic $2$

Let $k$ be an algebraically closed field, $S$ a variety over $k$ and m a nonnegative integer. There is a space $S_m$ over $S$ , called the jet scheme of $X$ of order $m$, parameterizing $m$-th jets on $S$. The fiber over the singular locus of $S$ is called the singular fiber. In this paper, we consider the singular fibers of the jet schemes of 2-dimensional rational double points over a field $k$ of characteristic $2$ whose resolution graph is of type $D_4$. There are two types of such singularities, of type $D_4^0$ and type $D_4^1$. We give the irreducible decomposition of the singular fiber and describe the configuration of the irreducible components. The case of a $D_4^0$-singularity is quite similar to the case of characteristic $0$ studied in [3]. The case of $D_4^1$-singularity requires more elaborate analysis of certain subsets of the singular fiber.