Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) canonical sheaf of the components?

In particular: to what extent the restriction of the dualizing sheaf of the global curve to a component can be described as the dualizing sheaf (possibly twisted by O(p), with $p$ attaching point) of that component?

There are, for instance, some examples that puzzle me. Take a smooth genus 3 curve, then its canonical model is a plane quartic. Then consider the curve given by a central elliptic curve attached to 2 elliptic "tails" (I am not sure this is the correct terminology). The "canonical model" of this should equally be a plane curve. What plane curve??