  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.
