Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Turning non-smooth points into rational points

Cesar Hilario

Abstract

In a recent paper, the author and Stöhr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth non-decomposed point on a regular geometrically integral curve into a rational point. In this note we improve this result, by establishing a new bound that is sharp in every characteristic $p > 0$.

Turning non-smooth points into rational points

Abstract

In a recent paper, the author and Stöhr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth non-decomposed point on a regular geometrically integral curve into a rational point. In this note we improve this result, by establishing a new bound that is sharp in every characteristic .
Paper Structure (6 sections, 8 theorems, 46 equations)

This paper contains 6 sections, 8 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Generalities on regular curves
  3. Regular curves
  4. Function fields
  5. The sharp bound
  6. Examples

Key Result

Theorem 1.1

Let $C|K$ be a regular curve and let $\mathfrak{p}\in C$ be a non-decomposed non-smooth point of geometric $\delta$-invariant $\delta(\mathfrak{p})>0$. Then the image $\mathfrak{p}_n \in C_n$ of $\mathfrak{p}$ is a $K$-rational point for all $n\geq \log_p(2\delta(\mathfrak{p}) + 1)$. If in addition

Theorems & Definitions (15)

  • Theorem 1.1: HiSt22
  • Theorem 1.2
  • Theorem 2.1: BedSt87
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 5 more