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 p which — seen as a prime of the separable function field K ( C ) | K — is non-decomposed in the base field extension K ‾ ⊗ K K ( C ) | K ‾ . In this paper we establish a bound for the number of iterated Frobenius pullbacks needed in order to transform p into a rational point. This provides an algorithm to compute geometric δ -invariants of non-smooth points and a procedure to construct fibrations with moving singularities of prescribed δ -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.