# n-canonical embedding

Let $C$ be a nodal curve and let $\omega$ be its dualizing sheaf. Let $n$ be a integer larger than 2. Does anyone knows how to show that $\omega^{\otimes n}$ separates points and tangent vectors? that is, to show $\omega^{\otimes n}$ provives an embedding in a projective space? If $C$ is a smooth curve, ok, we could use the Riemann-Roch for do it, but and in this case where $C$ is nodal? Does Anyone knows?

Everywhere I looked, assumes that fact, but displays no proof.

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Use Riemann-Roch for the normalization. – Will Sawin Apr 11 '13 at 22:32
@Artin009: In the absence of smoothness, is "separating points and tangent vectors" the right way to think about things? Anyway, have you read the proof of very ampleness for $n > 2$ in Theorem 1.2 of the Deligne-Mumford paper? (Please also consider to use a name other than "Artin009".) @Will: Since $\omega$ is not the pushforward of a line bundle on the normalization, why is RR on the normalization relevant? Isn't duality on the nodal curve a more appropriate argument? For example, that is what Deligne and Mumford do. – user30379 Apr 12 '13 at 0:37
"In the absence of smoothness, is "separating points and tangent vectors" the right way to think about things? "Well, I guess so. At least these notes MIT this makes sense, see exercise 3.1. ocw.mit.edu/courses/mathematics/… – Artin009 Apr 12 '13 at 1:42
@Artin009: Just because those MIT notes say it that way doesn't mean it is the right way to think about it (and those notes don't explain the exercise or the precise meaning of the terminology; curious that the notes mask all evidence of who wrote them, as far as I can tell). I recommend looking at my above suggested reference in Deligne-Mumford, where you'll see them carry out the actual cohomological computations using duality on the semistable curve, and I hope that should clarify the situation for you. – user30379 Apr 12 '13 at 4:29
Use the explicit description of the dualizing sheaf on the nodal curve: if $X$ is a nodal curve and $X'$ is its normalization, then (local) sections of $\omega_X$ are rational differentials $\omega$ on $X'$ satisfying the following three conditions. 1. They are regular outside inverse images of nodes. 2. They have poles of order at most one at the points mapping to nodes. 3. If $a,b\in X'$ are mapped to the same node of $X$, then $\mathrm{res}_a\omega+\mathrm{res}_b\omega=0$. – Serge Lvovski Apr 12 '13 at 7:17