The **gonality** (or **$k$-gonality**) of an integral algebraic curve $X$ over a field $k$ is the minimal degree of a dominant $k$-morphism $X \to \mathbb P^1_k$. Equivalently, it is the minimum of $[k(X):k(t)]$ as $t$ ranges over (transcendental) elements of the function field $k(X)$ of $X$.

The base extension of any dominant morphism $X \to \mathbb{P}^1_k$ is a dominant morphism $X_{\overline{k}} \to \mathbb{P}^1_{\overline{k}}$, so the geometric gonality of $X$ is less than or equal to the $k$-gonality of $X$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Bjorn Poonen on 2022-03-24 19:02:30

**Referred to by:**

- ag.gonality_geom
- columns.gps_gl2zhat_fine.q_gonality
- columns.gps_gl2zhat_fine.q_gonality_bounds
- columns.gps_gl2zhat_fine.qbar_gonality
- columns.gps_gl2zhat_fine.qbar_gonality_bounds
- modcurve.109.110.8.a.1.top
- modcurve.34.54.3.a.1.top
- modcurve.38.60.4.a.1.top
- modcurve.gonality
- modcurve.plane_model
- shimcurve.gonality

**History:**(expand/hide all)

- 2022-03-24 19:02:30 by Bjorn Poonen (Reviewed)
- 2022-03-24 18:59:38 by Bjorn Poonen
- 2022-03-21 21:27:19 by Bjorn Poonen
- 2022-03-20 16:50:01 by Andrew Sutherland

**Differences**(show/hide)