Let $\nu:\tilde{X}\to X$ be the normalization of a projective variety with non-isolated singularity. The usual object to consider is $\nu_*\mathcal{O}_{\tilde{X}}/\mathcal{O}_X$. For example, one would like to compute $\chi(\nu_*\mathcal{O}_{\tilde{X}}/\mathcal{O}_X)$. Or even particular cohomologies. (They all are finite, right?)

If the singularity is isolated everything works fine. In the non-isolated case this quotient is supported on the locus of "non-normality". It seems to be an unpleasant sheaf. At least, I am stuck even in the simplest case $\{x^d=y^{d-1}z\}\subset\Bbb P^3$.

Any general remarks on the computation/understanding of this quotient? References? Or at least how to compute for this particular example?