### Abstract

We show, 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^{2}_{1} < c_{2}.

Original language | English (US) |
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Pages (from-to) | 1791-1808 |

Number of pages | 18 |

Journal | New York Journal of Mathematics |

Volume | 23 |

State | Published - Dec 11 2017 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- General type
- Heights
- Hyperbolicity
- Moduli spaces
- Rational points
- Vojta’s conjecture

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## Cite this

Ascher, K., & Javanpeykar, A. (2017). Bounding heights uniformly in families of hyperbolic varieties.

*New York Journal of Mathematics*,*23*, 1791-1808.