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Karl-Otto Stöhr

Publications and source records attributed to Karl-Otto Stöhr.

5 recordsLinked to original sources
arXiv2025

Fibrations by plane quartic curves with a canonical moving singularity

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric generic fibre, which determines the generic behaviour of the special fibres, is an integral plane projective rational quartic curve over the algebraic closure of the function field of the base. It has the remarkable property that the tangent lines at the non-singular points are either all bitangents or all non-ordinary inflection tangents; moreover it is strange, that is, all the tangent lines meet in a common point. We construct two fibrations that are universal in the sense that any other fibration with the aforementioned properties can be obtained from one of them by a base extension. Furthermore, among these fibrations we choose a pencil of plane quartic curves and study in detail its geometry. We determine the corresponding minimal regular model and we describe it as a purely inseparable double covering of a quasi-elliptic fibration.

math.AG↗
arXiv2024

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 $δ$-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.

math.AG↗