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 $X$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #7349 by Tim Holzschuh on