For a singular stable curve $X$, its dualizing sheaf $\omega_X$ plays (to some extent) the role of the canonical bundle of a smooth curve. For instance, $\omega_X$ is ample, $h^0(\omega_X)=p_a(X)$ and has degree $2p_a(X)-2$, $p_a$ being the arithmetic genus. My questions are the following:
i) If we let $X_0$ be a general irreducible stable curve (i.e. you take a smooth curve of genus $g-1$ and identify two points) and we let $X_i$ be the nodal curve, which we get after attaching a curve of genus $g-i$ with a curve of genus $i$, for which of these singular curves is the dualizing sheaf base point free?
ii) In the cases when it is base point free, can we understand the map geometrically in the projective space $\mathbb{P}^{p_a(X)-1}$?
Thanks in advance.