---
tag: 6W8TDV9VVN
kind: concept
name: Genus of a curve
msc: [14H99, 14H50]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_genus
---

## Summary

The genus of a smooth projective connected curve is the dimension of its space of regular differential $1$-forms.
It is the main invariant of a curve.
Curves of genus $0$ are conics, and curves of genus $1$ with a rational point are elliptic curves.
Curves of genus at least $2$ have finitely many automorphisms.
Over $\C$ it equals the topological [[genus|NJ787GZS1N]] of the associated compact Riemann surface, the number of handles.
For a singular curve, one takes the genus of its normalization, called the geometric genus.

## Explanations

### L1 human

Over the complex numbers, a smooth projective connected curve is a closed surface, a sphere with some handles.
The number of handles is the genus.
The projective line is a sphere, of genus $0$.
A smooth plane cubic such as $y^2 = x^3 - x$ is a torus, of genus $1$.
The genus controls almost everything about the curve, for instance through the [[Riemann–Roch theorem|WMVAT9MBZ1]].