# Greatest common divisors of integral points of numerically equivalent divisors

## TL;DR

This paper extends G.C.D. results for integral points on algebraic varieties, showing that the height of certain subschemes remains small at integral points under specific divisor conditions, using advanced Diophantine approximation techniques.

## Contribution

It generalizes existing G.C.D. results to broader settings involving Cohen-Macaulay varieties and numerically equivalent divisors, employing a novel approach based on Schmidt's subspace theorem.

## Key findings

- Height of subschemes is small at integral points under divisor conditions
- Generalization of G.C.D. results to Cohen-Macaulay projective varieties
- Application of Ru--Vojta's subspace theorem in Diophantine geometry

## Abstract

We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb G_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen-Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate as an application of Vojta's conjecture, which is substituted by a more general version of Schmidt's subspace theorem of Ru--Vojta in our proof.