• Corpus ID: 53590030

Bounding heights uniformly in families of hyperbolic varieties

@article{Ascher2016BoundingHU,
  title={Bounding heights uniformly in families of hyperbolic varieties},
  author={Kenneth Ascher and Ariyan Javanpeykar},
  journal={arXiv: Algebraic Geometry},
  year={2016}
}
We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of Su-Ion Ih for curves of genus at least two to higher-dimensional varieties. As an application, we show that, assuming Vojta's height conjecture, the height of a rational point on a curve of general type is uniformly bounded. Finally, we prove a similar result for smooth hyperbolic surfaces with $c_1^2 > c_2$. 
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References

SHOWING 1-10 OF 48 REFERENCES

Height Uniformity for Algebraic Points on Curves

  • S. Ih
  • Mathematics
    Compositio Mathematica
  • 2002
We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of ‘Uniformity of rational points.’ It says that the Lang conjecture on the distribution of rational points on varieties…

Height uniformity for integral points on elliptic curves

We recall the result of D. Abramovich and its generalization by P. Pacelli on the uniformity for stably integral points on elliptic curves. It says that the Lang-Vojta conjecture on the distribution…

Lang's Conjectures, Fibered Powers, and Uniformity

We prove that the fibered power conjecture of Caporaso et al. (Con- jecture H, (CHM), §6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as…

Birational Geometry of Algebraic Varieties

Needless to say, tlie prototype of classification theory of varieties is tlie classical classification theory of algebraic surfaces by the Italian school, enriched by Zariski, Kodaira and others. Let…

Correlation for surfaces of general type

The main geometric result of this paper is that given any family of surfaces of general type f:X-->B, for sufficiently large n the fiber product X^n_B dominates a variety of general type. This result…

Theta height and Faltings height

Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized abelian variety. We also give…

Compact moduli spaces of surfaces of general type

We give an introduction to the compactification of the moduli space of surfaces of general type introduced by Koll\'ar and Shepherd-Barron and generalized to the case of surfaces with a divisor by…

Level structures on abelian varieties and Vojta’s conjecture

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$ , there is an integer $m_{0}$ such that for any…

Symmetric differentials and variations of Hodge structures

  • Yohan Brunebarbe
  • Mathematics
    Journal für die reine und angewandte Mathematik (Crelles Journal)
  • 2018
Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures…

On the negativity of kernels of Kodaira–Spencer maps on Hodge bundles and applications

Viehweg recently asked that if the cotangent bundles of moduli varieties of polarized varieties with log-pole along infinity are positive in some sense. It is well known that the moduli space of…