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In Hartshorne's book Algebraic Geometry he says in Example 3.3.5 in Chapter IV (p. 309):

If $X$ is a plane curve of degree 4, then $D=X.H$ is a very ample divisor of degree 4.

Here $H$ is a hyperplane section for a projective embedding. Why is this divisor very ample?

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It is very ample by definition: If $i:X \to \mathbb{P}^2$ is the projective embedding then $\mathcal{L}(D) = i^*\mathcal{O}_{\mathbb{P}^2}(1) = i^*\mathcal{O}_{\mathbb{P}^2}(H)$ is a very ample line bundle by (Hartshorne, p.120 Definition). By Hartshorne p.307 a divisor $D$ is called very ample if $\mathcal{L}(D)$ is very ample.

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  • $\begingroup$ Yes that makes a lot of sense, thank you! I was confusing myself by trying to prove this in a difficult way instead of just thinking about the definition. $\endgroup$
    – Misja
    Commented May 13, 2015 at 0:02

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