Very ample linear system

A very ample linear system defines an embedding into projective space. On a smooth projective curve this means it has no base points and separates both distinct points and tangent directions; the complete system of a very ample divisor is an example.

For a finite generating space $V\subseteq H^0(X,\mathcal{L})$ on a projective scheme, its map to projective space is a closed immersion exactly when, after algebraic closure of the ground field, the displayed evaluation is surjective for every length-two closed subscheme $E$. Two distinct points test point separation; a double point tests a direction in the Zariski tangent space. For the complete section space, this characterizes a very ample line bundle.