## A subspace theorem for subvarieties.(English)Zbl 1439.11181

Summary: 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 D. McKinnon and M. Roth [Invent. Math. 200, No. 2, 513–583 (2015; Zbl 1337.14023)]. The inequality obtained in this paper connects the result of McKinnon and Roth (the case when the subschemes are points) and the results of P. Corvaja and U. Zannier [Am. J. Math. 126, No. 5, 1033–1055 (2004; Zbl 1125.11022)], J.-H. Evertse and R. G. Ferretti [Int. Math. Res. Not. 2002, No. 25, 1295–1330 (2002; Zbl 1073.14521)], the author [Funct. Approximatio, Comment. Math. 56, No. 2, 143–163 (2017; Zbl 1432.11101)], and the author and P. Vojta [“A birational Nevanlinna constant and its consequences”, Preprint, arXiv:1608.05382] (the case when the subschemes are divisors). Furthermore, our approach gives an alternative short and simpler proof of McKinnon and Roth’s result.

### MSC:

 11J97 Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.) 11J87 Schmidt Subspace Theorem and applications 14G05 Rational points

### Citations:

Zbl 1337.14023; Zbl 1125.11022; Zbl 1073.14521; Zbl 1432.11101
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### References:

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