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

We generalize the gcd results of Corvaja and Zannier and of Levin on 𝔾mn 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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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
Received: 25 February 2020
Revised: 2 May 2020
Accepted: 2 July 2020
Published: 1 March 2021
Julie Tzu-Yueh Wang
Institute of Mathematics
Academia Sinica
Yu Yasufuku
Department of Mathematics, College of Science and Technology
Nihon University
Chiyoda-ku, Tokyo