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Let $X$ be a projective scheme over an algebraically closed field. It is a well-known theorem that a line bundle $L$ defines a closed embedding into projective space if and only if it separates points and tangent vectors, which means: (1) For any distinct closed points $P, Q \in X$ there is an $s \in \Gamma(X,L)$ with the property that $s\in m_PL_P$ but $s \not\in m_QL_Q$. (2) For every closed point $P \in X$ the set $\{s \in \Gamma(X,L)|s_P \in m_PL_P\}$ spans the vector space $m_PL_P/m_P^2L_P$.

Now in Hartshorne (Prop. IV 3.4.) this criterion is applied in the following situation: One wants to show that every curve $C$ can be embedded into $\mathbb{P}^3$. First one embedds the curve $C$ into an arbitrary $\mathbb{P}^n$. Then one chooses some $O \in \mathbb{P}^n$ andprojects the curve down from $O$ into $\mathbb{P}^{n-1}$. Then one wants to show that this projection map is a closed immersion if and only if (a) $O$ does not lie on any secant of $C$ (b) $O$ does not lie on any tangent of $C$. At this point Hartshorne applies the above criterion. But at this point I do not understand why the fact that $O$ does not lie on any tangent implies $(2)$ of the above criterion. I understood why $(a)$ implies $(1)$ but why does $(b)$ yield that the line bundle separates tangent vectors

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[I assume $C$ is smooth]

The point is this: $m_PL_P/m_P^2L_P$ is $1$-dimensional and hence generating it is equivalent to finding a single element that is not zero. The statement that all sections that are in $m_PL_P$ are also in $m_P^2L_P$ is equivalent with the statement that every member of the corresponding linear system that contains $P$, also contains the tangent line to $C$ at $P$. I think you should be able to finish from here.

[A modification of this idea actually works in higher dimensions, so the first sentence is not really necessary, but makes it easier in the curve case.]

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  • $\begingroup$ Thank you for your answer. But why is The statement that all sections that are in $m_PL_P$ are also in $m_P^2L_P equivalent with the statement that every member of the corresponding linear system that contains P, also contains the tangent line to C at P. I'm sorry, but this is not immediately clear for me. $\endgroup$ – phil Jul 12 '11 at 1:53
  • $\begingroup$ A section is in $m_P^2L_P$ if the corresponding divisor (in the ambient space) is tangent to the curve. In other words it contains the tangent line. Another way to see it is that the tangent line is the dual of $m_P/m_P^2$. $\endgroup$ – Sándor Kovács Jul 12 '11 at 8:08

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