# Divisorial instability and Vojta's Main Conjecture for $\mathbb{Q}$-Fano varieties

## TL;DR

This paper links divisorial instability of Fano varieties to Vojta's Main Conjecture, using concepts from K-stability, Newton-Okounkov bodies, and an arithmetic form of Cartan's Second Main Theorem.

## Contribution

It demonstrates that divisorial instability implies cases of Vojta's Main Conjecture for Fano varieties, connecting stability notions with Diophantine properties.

## Key findings

- Divisorial instability implies instances of Vojta's Main Conjecture.
- Interpretation of stability via Newton-Okounkov bodies.
- Application of an arithmetic form of Cartan's Second Main Theorem.

## Abstract

We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.