Greatest common divisors of integral points of numerically equivalent divisors

Wang, Julie Tzu-Yueh; Yasufuku, Yu

doi:10.2140/ant.2021.15.287

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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

DOI: 10.2140/ant.2021.15.287

Abstract

We generalize the gcd results of Corvaja and Zannier and of Levin on $𝔾_{m}^{n}$ 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

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

Vol. 15, No. 1, 2021