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Greatest common divisors of integral points of numerically equivalent divisors
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-03-01, DOI: 10.2140/ant.2021.15.287
Julie Tzu-Yueh Wang, Yu Yasufuku

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.