Lemma 33.23.2. Let $k$ be an algebraically closed field. Let $X$ be a proper $k$-scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Suppose that for every closed subscheme $Z \subset X$ of dimension $0$ and degree $2$ over $k$ the map

$H^0(X, \mathcal{L}) \longrightarrow H^0(Z, \mathcal{L}|_ Z)$

is surjective. Then $\mathcal{L}$ is very ample on $X$ over $k$.

Proof. This is a reformulation of Lemma 33.23.1. Namely, given distinct closed points $x, y \in X$ taking $Z = x \cup y$ (viewed as closed subscheme) we get condition (1) of the lemma. And given a nonzero tangent vector $\theta \in T_{X/k, x}$ the morphism $\theta : \mathop{\mathrm{Spec}}(k[\epsilon ]) \to X$ is a closed immersion. Setting $Z = \mathop{\mathrm{Im}}(\theta )$ we obtain condition (2) of the lemma. $\square$

There are also:

• 2 comment(s) on Section 33.23: Separating points and tangent vectors

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E8T. Beware of the difference between the letter 'O' and the digit '0'.