Bounding heights uniformly in families of hyperbolic varieties (Q1747853)

A conjecture due to S. Lang on rational points on varieties of general type implies a uniform bound, depending only on \(k\) and \(g\), of the number of \(k\)-points on a curve of general type and genus \(g\) over a number field \(k\) [\textit{L. Caporaso} et al., J. Am. Math. Soc. 10, No. 1, 1--35 (1997; Zbl 0872.14017)]. In [Compos. Math. 134, No. 1, 35--57 (2002; Zbl 1031.11041)], \textit{S.-i. Ih} has shown that Vojta's height conjecture [\textit{P. Vojta}, Int. Math. Res. Not. 1998, No. 21, 1103--1116 (1998; Zbl 0923.11059)] implies that the height of a rational point on a smooth proper curve of general type is bounded uniformly in families, with the bound depending linearly on the height of the curve. In [\textit{S.-i. Ih}, Trans. Am. Math. Soc. 358, No. 4, 1657--1675 (2006; Zbl 1118.14026)] \textit{S.-i. Ih} showed that the same is true for the integral points on elliptic curves. In the paper under review, the authors generalize Ih's results to higher dimensional varieties by investigating consequences of Vojta's conjecture for families of hyperbolic varieties of general type. The authors also explain why stronger uniformity type statements on varieties of general type cannot be expected. The proof uses the recent paper [\textit{D. Abramovich} and \textit{A. Várilly-Alvarado}, Adv. Math. 329, 523--540 (2018; Zbl 1405.14115)] which shows that Vojta's Conjecture implies a version of the conjecture for stacks. As an application, the authors show that, assuming Vojta's height conjecture, for \(g\geq 2\) an integer and \(k\) a number field, there is a real number \(c(g,k)\) depending only on \(g\) and \(k\) such that, for all smooth projective curves \(X\) of genus \(g\) over \(k\) and for all \(P\) in \(X(k)\), we have \[ h(P)\leq c(g,k) (h(X)+d_k({\mathcal{T_X}})). \] Here, \(d_k({\mathcal{T}})\) is the relative discriminant of a stack over a number field.