GCD inequalities arising from codimension‐2 blowups

Abstract Assuming a deep Diophantine geometry conjecture by Vojta, Silverman proved an inequality giving an upper bound for the greatest common divisor (GCD). In this paper, we unconditionally prove a weaker version of this inequality. The main ingredient is the Ru–Vojta theory, which provides an efficient method of using Schmidt subspace theorem. The proof strategy is similar to a previous work by the author, but since we work with blowups along a codimension‐2 subvariety instead of blowups along points, algebro‐geometric inputs such as computing intersection numbers and giving a criterion for ampleness become more complicated. The same method also proves an upper bound for the simultaneous counting function in Nevanlinna theory.