Definition:Genus of Proper One-Dimensional Scheme

Definition
Let $k$ be a field.

Let $\struct {X, \OO_X}$ be a scheme over $k$ of Krull dimension $1$ and proper over $k$.

Let $\map {H^i} {X, \OO_X}$ denote the $i$-th sheaf cohomology.

Let $\dim_k \map {H^0} {X, \OO_X} = 1$.

Then the genus of $X$ is defined as:
 * $g := \dim_k \map {H^1} {X, \OO_X}$