On the normalization and the quotient of the structure sheaves

Question

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?

Answer 1

The cohomologies are all finite since the quotient is coherent. Macaulay2 should be able to give you a presentation of that module.

For example, you can use the integralClosure command (now a package which is loaded by default) to compute the integral closure $S$ of a ring $R$. It is then easy to view that ring as an $R$-module if it is homogeneous for some appropriate weighting (your example above can be weighted to make it homogeneous I think, $\text{deg} x = d-1, \text{deg} y = d, \text{deg} z = d(d-1)$), or you can probably use the innards for the IntegralClosure package as well (see the conductor command which computes the annihilator of the module you define above).