On regular but non-smooth integral curves

Let $C$ be a regular geometrically integral curve over an imperfect field $K$ and assume that it admits a non-smooth point $\mathfrak{p}$ which -- seen as a prime of the separable function field $K(C)|K$ -- is non-decomposed in the base field extension $\overline{K} \otimes_K K(C)|\overline{K}$. In this paper we establish a bound for the number of iterated Frobenius pullbacks needed in order to transform $\mathfrak{p}$ into a rational point. This provides an algorithm to compute geometric $\delta$-invariants of non-smooth points and a procedure to construct fibrations with moving singularities of prescribed $\delta$-invariants. We show that the bound is sharp in characteristic 2. We further study the geometry of a pencil of plane projective rational quartics in characteristic 2 whose generic fibre attains our bound. On our way, we prove several results on separable and non-decomposed points that might be of independent interest.