# Relative canonical divisor associated to toric morphism induced by refinement of fan

Let $\phi:X'\to X$ be a morphism between toric varieties $X=X(\Delta), X'=X'(\Delta')$, induced by a refinement $\Delta'$ of $\Delta$. This refinement is obtained from a sequence of stellar subdivisions, and hence the exceptional locus of $\phi$ is a divisor.

We do not assume that $\Delta$ or $\Delta'$ are smooth, but they are Gorenstein.

Can anything be said about the relative canonical divisor $K_{X'/X}$?

Some more details: $K_{X'/X}$ is, as always, supported on the exceptional locus of $\phi$. This exceptional locus is the union of the divisorial torus orbits corresponding to the new rays created during the stellar subdivision. So if $\rho_i$ are the new rows, then $K_{X'/X}=\sum_i k_iD_{\rho_i}$ for some $k_i\in\mathbb{Z}$. I am particularly interested in a description of the $k_i$ in terms of the fans $\Delta,\Delta'$.

• If someone tells me what's unclear about the question as is, then I'll edit it to make it more clear – user2520938 May 13 '17 at 20:05

You can say that $K_{X'/X}$ is an effective Cartier divisor. This is because toric varieties have rational singularities, which together with being Gorenstein implies that $X$ and $X'$ have canonical singularities which in this case means exactly that $K_{X'/X}$ is effective. It is Cartier because of the Gorenstein assumption. I don't think you can say much more.
• In my question I don't make any assumptions on $X$ and $X'$ other then them being Gorenstein, so I'm not sure what you mean with your first statement? – user2520938 May 18 '17 at 17:20
• You said $\Delta$ and $\Delta'$ were Gorenstein. I assumed that this means that $X$ and $X'$ are, but wasn't sure. Cheers. – Sándor Kovács May 18 '17 at 17:23