Locally free sheaf

A sheaf of ${\mathcal{O}}_X$-modules is locally free of finite rank when every point has an open neighborhood $U$ on which it is isomorphic to ${\mathcal{O}}_U^{\oplus r}$ for some finite $r$. If $X$ is connected, the rank is constant.

A line bundle on a scheme is a locally free sheaf of rank one. Its global sections can define a morphism to projective space when they have no common zero.

The Picard group is the group of isomorphism classes of line bundles under tensor product.

A line bundle is very ample when its global sections define a closed embedding into projective space.

A basepoint-free vector space of global sections $V\subseteq H^0(X,L)$ defines the Kodaira map $X\to \mathbb{P}(V^{\ast })$ by evaluating the sections at each point.