  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.
