## Experimental Mathematics

- Experiment. Math.
- Volume 18, Issue 3 (2009), 347-367.

### The Geometric Bogomolov Conjecture for Curves of Small Genus

#### Abstract

The Bogomolov conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov conjecture for all curves of genus at most $4$ over a function field of characteristic zero. We recover the known result for genus-$2$ curves and in many cases improve upon the known bound for genus-$3$ curves. For many curves of genus $4$ with bad reduction, the conjecture was previously unproved.

#### Article information

**Source**

Experiment. Math., Volume 18, Issue 3 (2009), 347-367.

**Dates**

First available in Project Euclid: 25 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1259158471

**Mathematical Reviews number (MathSciNet)**

MR2555704

**Zentralblatt MATH identifier**

1186.11035

**Subjects**

Primary: 11G30: Curves of arbitrary genus or genus = 1 over global fields [See also 14H25]

Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 11G50: Heights [See also 14G40, 37P30]

**Keywords**

Bogomolov conjecture curves of higher genus function fields metric graphs

#### Citation

Faber, X. W. C. The Geometric Bogomolov Conjecture for Curves of Small Genus. Experiment. Math. 18 (2009), no. 3, 347--367. https://projecteuclid.org/euclid.em/1259158471