The Stacks project

Definition 53.3.1. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $d \geq 0$ and $r \geq 0$. A linear series of degree $d$ and dimension $r$ is a pair $(\mathcal{L}, V)$ where $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module of degree $d$ (Varieties, Definition 33.44.1) and $V \subset H^0(X, \mathcal{L})$ is a $k$-subvector space of dimension $r + 1$. We will abbreviate this by saying $(\mathcal{L}, V)$ is a $\mathfrak g^ r_ d$ on $X$.