• 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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