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An upper bound for the generalized greatest common divisor of rational points

Benjamín Barrios

Abstract

Let $X$ be a smooth projective variety defined over a number field $K$. We give an upper bound for the generalized greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over $K$. To prove the result, we stablish a rather uniform Riemann--Roch type inequality.

An upper bound for the generalized greatest common divisor of rational points

Abstract

Let be a smooth projective variety defined over a number field . We give an upper bound for the generalized greatest common divisor of a point with respect to an irreducible subvariety also defined over . To prove the result, we stablish a rather uniform Riemann--Roch type inequality.

Paper Structure

This paper contains 4 sections, 7 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Riemann--Roch
  3. Proof of the GCD bound
  4. Acknowledgments

Key Result

Theorem 1.1

Let $X$ be a smooth projective variety defined over a number field $K$ of dimension $n$ and $\mathcal{A}$ be an ample line sheaf on $X$. Let $Y$ be a reduced closed sub-scheme consisting on $d$ geometric points. Then given any $\varepsilon>0$ there is a properly contained Zariski closed set $Z_\vare as $x$ varies in $(X-Z_{\varepsilon})(\overline{K})$.

Theorems & Definitions (12)

  • Definition 1: MR2162351*Definition 2
  • Theorem 1.1: MR4756363*Theorem 3.1
  • Theorem 1.2: GCD bound
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 2 more