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