On essentially large divisors

TL;DR

This paper introduces the concept of essentially large divisors, exploring their geometric and arithmetic implications, and proves that certain divisors on specific varieties are essentially large, extending classical theorems.

Contribution

It defines essentially large divisors and establishes conditions under which divisors on certain varieties are essentially large, generalizing classical results.

Findings

Effective divisors with many components in general position are essentially large on certain varieties.

The notion of essentially large divisors links geometric configurations to arithmetic properties.

Generalizes classical theorems of Picard and Siegel to broader divisor contexts.

Abstract

Motivated by the classical Theorems of Picard and Siegel and their generalizations, we define the notion of an {\it essentially large} effective divisor and derive some of its geometric and arithmetic consequences. We then prove that on a nonsingular projective variety $X$ whose codimension is no greater than $\dim X-2$, every effective divisor with $\dim X +2$ or more components in general position is essentially large.