arXiv subjects

Jakab Schrettner

Publications and source records attributed to Jakab Schrettner.

3 recordsLinked to original sources

arXiv2026

Defining reduction types of curves via minimal regular and minimal normal crossings models

We propose a definition of the reduction type of a curve over a discretely valued field in terms of the special fibre of an arbitrary regular model. We show that under this definition, the reduction type in terms of the minimal regular model determines and reduction type in terms of the minimal regular normal crossings model and vice versa. We also show that our definition is compatible with earlier classification results in low genus, namely the Kodaira-N\'eron classification for elliptic curves and the classification result of Namikawa-Ueno in genus 2. The essential new element in our definition is a suitable invariant of the singularities on the special fibre, building on the notion of equisingularity introduced by Zariski.

math.NT↗

arXiv2025

Local constancy of reduction type and related invariants for curves in $p$-adic families

We investigate the behaviour of the reduction type and related invariants of curves in families of curves over a discretely valued field. By a family, we will mean a set of curves obtained by perturbing the coefficients of the defining equations. We will show that the reduction type in these families is locally constant in the topology induced by the valuation. We also derive local constancy results for some related invariants, such as the Tamagawa number, the Birch and Swinnerton-Dyer 'fudge factor' and the Galois representation.

math.NT↗

arXiv2020

Avoiding right angles and certain Hamming distances

In this paper we show that the largest possible size of a subset of $\mathbb{F}_q^n$ avoiding right angles, that is, distinct vectors $x,y,z$ such that $x-z$ and $y-z$ are perpendicular to each other is at most $O(n^{q-2})$. This improves on the previously best known bound due to Naslund \cite{Naslund} and refutes a conjecture of Ge and Shangguan \cite{Ge}. A lower bound of $n^{q/3}$ is also presented. It is also shown that a subset of $\mathbb{F}_q^n$ avoiding triangles with all right angles can have size at most $O(n^{2q-2})$. Furthermore, asymptotically tight bounds are given for the largest possible size of a subset $A\subseteq \mathbb{F}_q^n$ for which $x-y$ is not self-orthogonal for any distinct $x,y\in A$. The exact answer is determined for $q=3$ and $n\equiv 2\pmod {3}$. Our methods can also be used to bound the maximum possible size of a binary code where no two codewords have Hamming distance divisible by a fixed prime $q$. Our lower- and upper bounds are asymptotically tight and both are sharp in infinitely many cases.

math.CO↗