#### 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.

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##### 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
##### Milestones
Received: 25 February 2020
Revised: 2 May 2020
Accepted: 2 July 2020
Published: 1 March 2021
##### Authors
 Julie Tzu-Yueh Wang Institute of Mathematics Academia Sinica Taipei Taiwan Yu Yasufuku Department of Mathematics, College of Science and Technology Nihon University Chiyoda-ku, Tokyo Japan