Dualizing sheaf of reducible variety? Sorry for my poor English.
Let $X$ be a reducible projective variety. 
My question is:


*

*How can I compute the dualizing sheaf of $X$ and express it in an explicit way?

*Is there a method to get dualizing sheaf of whole reducible variety $X$ from the information of dualizing sheaves of its irreducible components?


Currently I'm not concern the general case, but I want a few accessible concrete examples such as:


*

*reducible hypersurfaces,

*union of toric varieties glued at isomorphic orbits(Alexeev calls it stable toric variety).


The reason why I concern is to understand the limit in moduli spaces of stable pairs. 
 A: For hypersurfaces the answer is as follows. Assume that you have components $X_i$ of a reducible hypersurface $X$. Then $\omega_X|_{X_i} \cong \omega_{X_i}(\sum_{i \neq j} X_j ) := \omega_{X_i} \otimes \mathcal{O}(\sum_{i \neq j} X_j )$. 
As for the proof: let $X':= \sum_{i \neq j} X_j$. Then we have a few equations:
$\omega_{X_i} \cong \omega_{P^n}(X_i)|_{X_i}$, $\omega_{X'} \cong \omega_{P^n}(X')|_{X'}$
So:
$\omega_{X_i}(X') \cong \omega_{P^n}(X_i+ X')|_{X_i} \cong  \left( \omega_{P^n} (X)  |_X \right) $ $|_{X_i}  \cong {\omega_X} |_{X_i}$
Concerning how you can get generally the canonical sheaf of a reduced scheme from its component, here is something very similar: if your scheme is semi-normal and $S_2$ (which is true for stable pairs), then when you take its normalization $\pi : \tilde{X} \to x$, the support of the points where $\pi$ is not an isomorphism, is a divisor, say $B$. Then you have the formula $\pi^* \omega_X \cong \omega_{\tilde{X}}(B)$. There is a little bit about this in the 4th section of http://arxiv.org/abs/0801.1541 .
A: With regards part 2.  
Let's assume that you have two components $X_1$ and $X_2$ (or even unions of components) such that $X_1 \cup X_2 = X$=.  Let $I_1$ and $I_2$ denote the ideal sheaves of $X_1$ and $X_2$ in $X$.
Set $Z$ to be the scheme $X_1 \cap X_2$, in other words, the ideal sheaf of $Z$ is $I_1 + I_2$. 
It is easy to see you have a short exact sequence
$$0 \to I_1 \cap I_2 \to I_1 \oplus I_2 \to (I_1 + I_2) \to 0$$
where the third map sends $(a,b)$ to $a-b$.
The nine-lemma should imply that you have a short exact sequence
$$0 \to O_X \to O_{X_1} \oplus O_{X_2} \to O_Z \to 0$$
If you Hom this sequence into the dualizing complex of $X$, you get a triangle
$$\omega_Z^. \to \omega_{X_1}^. \oplus \omega_{X_2}^. \to \omega_{X}^. \to \omega_Z^.[1]$$
You can then take cohomology and, depending on how things intersect (and what you understand about the intersection), possibly answer your question.  
If $X_1$ and $X_2$ are hypersurfaces with no common components (which should imply everything in sight is Cohen-Macualay) then these dualizing complexes are all just sheaves (with various shifts), and you just get a short exact sequence
$$0 \to \omega_{X_1} \oplus \omega_{X_2} \to \omega_{X} \to \omega_{Z} \to 0$$
Technically speaking, I should also probably push all these sheaves forward onto $X$ via inclusion maps.
