A generalized Schmidt subspace theorem for closed subschemes

We prove a generalized version of Schmidt's subspace theorem for closed subschemes in general position terms suitably defined Seshadri constants with respect to fixed ample divisor. Our proof builds on previous work Evertse and Ferretti, Corvaja Zannier, others, uses standard techniques from algebraic geometry such as notions positivity, blowing-ups direct image sheaves. As an application, we recover higher-dimensional Diophantine approximation K.~F.~Roth-type due D.~McKinnon M.~Roth significantly shortened proof, while simultaneously extending the scope use this context natural way.