Length-two criterion for a very ample linear system ($V\twoheadrightarrow H^0(E,\mathcal{L}{|}_E),\mathrm{length}E=2$)

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.