Vol. 15, No. 1, 2021

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Greatest common divisors of integral points of numerically equivalent divisors

Julie Tzu-Yueh Wang and Yu Yasufuku

Vol. 15 (2021), No. 1, 287–305
Abstract

We generalize the gcd results of Corvaja and Zannier and of Levin on ${\mathbb{𝔾}}_{m}^{n}$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen–Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman’s gcd estimate, but instead of his usage of Vojta’s conjecture, we use the recent result of Ru and Vojta.

Keywords
height inequality, integral points, greatest common divisors, blowups, Schmidt subspace theorem, Vojta's conjecture, entire curves
Mathematical Subject Classification 2010
Primary: 11J97
Secondary: 11J87, 14G05, 32A22