= Length-two criterion for a very ample linear system
{title2=$V\twoheadrightarrow H^0(E,\mathcal L|_E),\quad\operatorname{length}E=2$}
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>.
Back to article page
