Let $Y$ be a complete intersection in a complete simplicial toric variety $X_\Sigma$ such that $\DeclareMathOperator{Sing}{Sing}\Sing(Y)\subset\Sing(X_\Sigma)$. Suppose that $\phi:X_{\widehat{\Sigma}}\to X_\Sigma$ is a toric resolution induced by a refinement $\widehat{\Sigma}$ of the fan $\Sigma$.
Is there a simple way to check if the restriction of $\phi$ to $\widehat{Y}=\phi^{-1}(Y)$ induces a crepant morphism $\widehat{Y}\to Y$?
If the subvariety $Y$ is transversal to all the strata, which is the case, for example, for generic complete intersections of base point free linear systems, then it is easy.
The restriction of the resolution to Y is crepant if and only if Y misses all strata that are images of exceptional divisors with nonzero discrepancy. Recall that exceptional divisors corresponds to the new rays in the refinement $\widehat \Sigma$, and the strata in question correspond to the minimum cones of $\Sigma$ that contain them.
For instance, in cases of hypersurfaces, we really don't care about the new rays of $\widehat \Sigma$ which lie in the interior of the maximum cones of $\Sigma$.
