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.

`$\omega_X$`

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