\begin{definition}[Definition:Genus of Proper One-Dimensional Scheme]
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:
:$g := \dim_k \map {H^1} {X, \OO_X}$
Category:Definitions/Algebraic Geometry
Category:Definitions/Schemes
\end{definition}