We propose a reformulation of the Faltings-Wüstholz nonlinear version of Schmidt's subspace theorem with the help of toric deformations and Chow polytopes. Moreover, we show that the arithmetic Bézout theorem in Arakelov geometry can be used to obtain a Bézout theorem for Mumford's degree of contact. This is a birational invariant often considered in geometric invariant theory (GIT). The originality of this last result relies on the interpretation of GIT as a degeneration of Arakelov geometry. This should enable us to transfer all known results of Arakelov geometry into GIT.
"Diophantine approximations and toric deformations." Duke Math. J. 118 (3) 493 - 522, 15 June 2003. https://doi.org/10.1215/S0012-7094-03-11834-7