In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in Gmn with algebraic subgroups of dimension n-2. Actually, the present conclusion will give more uniform bounds with respect to the former statement. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of 'unlikely intersections'.
Capuano, L., Masser, D., Pila, J., Zannier, U. (2016). Rational points on Grassmannians and unlikely intersections in tori. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 48(1), 141-154 [10.1112/blms/bdv091].