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Effective calculation of local Weil functions via presentations of Cartier divisors

Nathan Grieve

TL;DR

The article develops an effective framework for computing local Weil functions on geometrically integral projective varieties by using presentations of Cartier divisors. It integrates effective Hilbert's Nullstellensatz, boundedness concepts with respect to fields of definition, and graded-module techniques to express local heights in concrete, computable terms. The main result, Theorem explicit:local:weil:constant, provides an effectively computable bound on the difference between local Weil functions associated to different presentations, paving the way for practical algorithms. This approach connects defining equations, syzygies, and Castelnuovo–Mumford regularity to Diophantine arithmetic, offering a foundational step toward algorithmic arithmetic geometry on projective varieties.

Abstract

We address the question of effectivity for calculation of local Weil functions from the viewpoint of presentations of Cartier divisors. This builds on the approach of Bombieri and Gubler as well as the perspective of our earlier works. Among other features, our approach here gives rise to theoretical effective algorithms for calculating local Weil functions on projective varieties.

Effective calculation of local Weil functions via presentations of Cartier divisors

TL;DR

The article develops an effective framework for computing local Weil functions on geometrically integral projective varieties by using presentations of Cartier divisors. It integrates effective Hilbert's Nullstellensatz, boundedness concepts with respect to fields of definition, and graded-module techniques to express local heights in concrete, computable terms. The main result, Theorem explicit:local:weil:constant, provides an effectively computable bound on the difference between local Weil functions associated to different presentations, paving the way for practical algorithms. This approach connects defining equations, syzygies, and Castelnuovo–Mumford regularity to Diophantine arithmetic, offering a foundational step toward algorithmic arithmetic geometry on projective varieties.

Abstract

We address the question of effectivity for calculation of local Weil functions from the viewpoint of presentations of Cartier divisors. This builds on the approach of Bombieri and Gubler as well as the perspective of our earlier works. Among other features, our approach here gives rise to theoretical effective algorithms for calculating local Weil functions on projective varieties.
Paper Structure (6 sections, 5 theorems, 121 equations)

This paper contains 6 sections, 5 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Selected conventions and other preliminaries
  3. Presentations of Cartier divisors
  4. Selected recall about effective Hilbert's Nullstellensatz
  5. Preliminaries about bounded subsets with respect to fields of definition
  6. Effective calculations of local Weil functions

Key Result

Lemma 3.1

Let $X$ be a geometrically integral projective variety defined over $\mathbf{K}$. Let $D$ be a Cartier divisor on $X$ and defined over a finite extension $\mathbf{F} / \mathbf{K}$ of $\mathbf{K}$. Then $X$ admits globally generated line bundles $L$ and $M$ which defined over $\mathbf{F}$ and which a

Theorems & Definitions (18)

  • Lemma 3.1: Compare with Hart
  • proof
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5
  • Example 3.6
  • Example 3.7
  • Example 3.8
  • Theorem 4.1: Jelonek:2005
  • ...and 8 more