Bertini and Northcott

TL;DR

This paper establishes a new Bertini-type theorem with explicit parameters, enabling better control over algebraic curves and facilitating the reduction of height and rank estimates on abelian varieties to Jacobian cases over number fields.

Contribution

It introduces a Bertini-type theorem with explicit control of geometric and arithmetic invariants, and offers a strategy to relate height and rank estimates on abelian varieties to Jacobians over extensions.

Findings

New Bertini-type theorem with explicit control of genus, degree, height, and field of definition.

A general strategy to reduce height and rank estimates on abelian varieties to Jacobian varieties.

Enhanced understanding of the arithmetic of algebraic curves and abelian varieties over number fields.

Abstract

We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field $K$ to the case of jacobian varieties defined over a suitable extension of $K$.