Geometric genus

The geometric genus of a smooth connected curve $ C$ is what one expects and can be defined in many ways, e.g. $ \frac{1}{2}\mathrm{dim} H^1(C)$. For an irreducible, singular curve, the geometric genus is then defined to be the genus of its normalization (the curve obtained by ungluing all the nodes).

Jeffrey Herschel Giansiracusa 2005-06-27