Primitive arcs on curves

Abstract

We introduce the notion of \emph{primitive arc} of a curve defined over a field $k$ and study criterions for the existence of such objects in terms of the geometry of the curve. We prove that this notion provides a criterion which determines when the normalization of an irreductible curve singularity $(X,x)$ induces an isomorphism between the formal neighborhoods of the associated arc schemes at the constant arc $x$ and its lifting $\bar x$ to the normalization $\bar X$. We also show that the existence of a primitive arc at $x\in X$ is equivalent to the smoothness of the analytically irreducible curve $X$ at $x$. In this end, we interpret this notion in terms of the formal deformations of the constant arc $x$ in the associated arc scheme.