Genus of a curve

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 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.