A subspace theorem for subvarieties

Abstract

We establish a height inequality, in terms of an (ample) line bundle, for a sum of subschemes located in $\ell$-subgeneral position in an algebraic variety, which extends a result of McKinnon and Roth (2015). The inequality obtained in this paper connects the result of McKinnon and Roth (the case when the subschemes are points) and the results of Corvaja and Zannier (2004), Evertse and Ferretti (2008), Ru (2017), and Ru and Vojta (2016) (the case when the subschemes are divisors). Furthermore, our approach gives an alternative short and simpler proof of McKinnon and Roth’s result.