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Generalized greatest common divisors, divisibility sequences, and Vojta’s conjecture for blowups. (English) Zbl 1197.11070
Summary: We apply Vojta’s conjecture to blowups and deduce a number of deep statements regarding (generalized) greatest common divisors on varieties, in particular on projective space and on abelian varieties. Special cases of these statements generalize earlier results and conjectures. We also discuss the relationship between generalized greatest common divisors and the divisibility sequences attached to algebraic groups, and we apply Vojta’s conjecture to obtain a strong bound on the divisibility sequences attached to abelian varieties of dimension at least two.

MSC:
11G35 Varieties over global fields
11D75 Diophantine inequalities
11J25 Diophantine inequalities
14G25 Global ground fields in algebraic geometry
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